Notes related to “Operators in quantum mechanics”

Armin Scrinzi

July 11, 2017

USE WITH CAUTION

These notes are compilation of my “scribbles” (only SCRIBBLES, although typeset in LaTeX). Thereabsolutely no time to unify notation, correct errors, proof-read, and the like. Your primary source mustby your own notes.

For now, it is only an outline to list a few of the topics that will be covered. The notes will be partiallybe filled in as the lecture proceeds.

Contents summary

The lecture consist of three main parts:

• the abstract motivation of quantum mechanics (C* algebras, GNS construction),

• key elements of the theory of unbounded operators (spectral theorem, Stone theorem), and

• elements of scattering theory (Møller operators, proof of asymptotic completeness).

Essential functional analysis will be introduced and selected topics will be treated in full mathematicalrigor. The course is intended for physicist with strong mathematical inclinations, but contact to theepistemological concepts of physics will be sought.

Purpose of the lecture

Quantum physics has spawned extensive use of functional analysis, but students in their majority areleft with little basic knowledge of the subject. A solid foundation of many techniques that are blindlyused by most physicists typically takes a 2-3 semester course in mathematics, which still may miss someadvanced issues that are relevant in practice. In addition, in a mathematics course usually little attentionis paid to the connection to the original motivation, the formation of intuitive pictures, and the actualimportance of certain formal questions.

This lecture is intended for physicist with strong mathematical inclinations. In a few points, conceptswill be introduced with mathematical rigor, which should put you into the position to at least see themajor inaccuracies committed in the remaining topics. In this spirit, an overview of many mathematicalquestions arising in quantum mechanics will be given and at least sketches of solutions and ideas of proofswill be provided.

Literature

The course of the lecture is strongly inspired byW. Thirring - Lehrbuch der Mathematischen Physik

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The part on unbounded operators follows closelyG. Teschl - Mathematical Methods in Quantum Mechanics

For basic reference we will largely useReed/Simon - Methods of Modern Mathematical Physics

Further literature will we given for selected topics.

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Contents

1 The C∗-algebra approach to classical and quantum physics 81.1 An analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2 C∗-algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Classical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.1 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.2 States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Justification of the C∗ algebraic nature of observables . . . . . . . . . . . . . . . . . . . . 101.4.1 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.2 State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.3 Normalization and positivity of a state . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.4 Equality of observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.5 C∗-structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.6 Basic structure of a physical theory . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 C∗ algebra 132.1 Banach space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Comment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Classes of elements of a C∗ algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Open set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4 The spectrum of A ∈ A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.1 Series expansion of the resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.2 The resolvent set is an open set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.3 Algebra element properties and spectrum . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.6 State on a C∗ algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.6.2 The states form a convex set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.6.3 Pure and mixed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Gelfand isomorphism 223.1 Weak *-topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Gelfand isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.1 Terms: Closure and dense sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.2 The spectral theorem in nuce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Representations in Hilbert space 264.1 Operator norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2 Representation of an algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2.1 Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.3 Irreducible representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.4 Sum and tensor product of representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.4.1 Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.5 Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 The GNS construction 295.1 Reducible and irreducible GNS representations . . . . . . . . . . . . . . . . . . . . . . . . 30

5.1.1 1 ∈ A/N is cyclic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.1.2 Irrep ⇔ pure state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.1.3 Irreps for abelian C∗ algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.1.4 Any representation is the (possibly infinite) sum of irreps . . . . . . . . . . . . . . 31

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6 Towards the spectral theorem for bounded operators 33

7 Elements of measure theory 337.1 Σ-algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.2 Borel sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.3 Measure and measurable space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.4 A few definitions for the Lebesgue integral (reminder) . . . . . . . . . . . . . . . . . . . . 347.5 Lebesgue measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.6 Absolutely continuous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7.6.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.7 Signed measures and complex valued measures . . . . . . . . . . . . . . . . . . . . . . . . 367.8 Projector valued measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.9 Positive operator valued measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.10 Measures and continuous functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

8 The spectral theorem for normal operators 378.1 Gel’fand isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.2 A state defines a measure on σ(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.3 Construction of the L2(dµi, σ(A)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.4 Spectral theorem by projection valued measures . . . . . . . . . . . . . . . . . . . . . . . . 388.5 Quantum habits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

9 Bounded and unbounded operators in Hilbert space 419.1 Linear operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.2 B.L.T. theorem (bounded line transformation) . . . . . . . . . . . . . . . . . . . . . . . . 41

9.2.1 Multiplication operator on L2(dx,R) . . . . . . . . . . . . . . . . . . . . . . . . . . 419.3 Comments on the domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.4 Quadratic form and polarization identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

9.4.1 Hermitian operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.5 Adjoint operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

9.5.1 Comment on the domain D(A∗) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.6 Range and kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.7 Self-adjoint operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

9.7.1 Multiplication operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.8 Closed operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.9 Normal operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

9.9.1 The domain of self-adjoint operators is maximal . . . . . . . . . . . . . . . . . . . 459.9.2 Closure of a symmetric operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469.9.3 The importance of self-adjointness for quantum mechanics . . . . . . . . . . . . . . 469.9.4 Example: the spatial derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469.9.5 A criterion for self-adjointness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489.9.6 Spectrum of a self-adjoint operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

9.10 Quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

10 The spectral theorem for self-adjoint operators 5010.1 Bounded spectral theorem with projection valued measure on R . . . . . . . . . . . . . . . 5010.2 Resolution of identity P (λ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.3 Construction of operators from projection valued measures . . . . . . . . . . . . . . . . . 51

10.3.1 The subspace Hψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5310.3.2 Unitarity, cyclic vectors, spectral basis . . . . . . . . . . . . . . . . . . . . . . . . . 53

10.4 For every projection valued measure there is a self-adjoint operator . . . . . . . . . . . . . 5410.5 For every self-adjoint operator there is a spectral measure . . . . . . . . . . . . . . . . . . 54

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10.5.1 The resolvent of P (a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5410.5.2 A measure is uniquely defined by its Borel transform . . . . . . . . . . . . . . . . . 5410.5.3 The resolvent defines a measure dµψ . . . . . . . . . . . . . . . . . . . . . . . . . . 5510.5.4 A resolution of identity P (λ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5610.5.5 The spectral projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5610.5.6 Spectral theorem for self-adjoint operators . . . . . . . . . . . . . . . . . . . . . . . 56

11 Stone’s theorem 5611.1 A self-adjoint operator generates a unitary group . . . . . . . . . . . . . . . . . . . . . . . 5611.2 Stone theorem - unitary groups derive from self-adjoint operators . . . . . . . . . . . . . . 57

12 Further important topics 5912.1 Self-adjoint Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5912.2 Self-adjointness of typical Hamiltonians in Physics . . . . . . . . . . . . . . . . . . . . . . 60

12.2.1 Domain of essential self-adjointness . . . . . . . . . . . . . . . . . . . . . . . . . . . 6012.2.2 Self-adjoint −∆ + V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

12.3 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6112.4 Analyticity of eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

12.4.1 Riesz projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

13 Scattering theory 6413.1 Scattering is our window into the microscopic world . . . . . . . . . . . . . . . . . . . . . 64

13.1.1 Cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6413.1.2 Potential scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

13.2 Mathematical modeling of a (simple) scattering experiment . . . . . . . . . . . . . . . . . 6413.3 The Møller operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

13.3.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6513.4 The Lippmann-Schwinger (LS) equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

13.4.1 In- and Out-states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6613.5 Spectral properties and dynamical behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 66

13.5.1 Spectral subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6613.5.2 Fourier transform of smooth functions (Riemann-Lebesgue) . . . . . . . . . . . . . 6813.5.3 Compact operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6813.5.4 The RAGE theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

13.6 The Møller operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7113.7 S-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

13.7.1 Asymptotic completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7213.7.2 Scattering operators compare two dynamics . . . . . . . . . . . . . . . . . . . . . . 7313.7.3 Intertwining property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

13.8 Asymptotic completeness - guide through the proof by Enss . . . . . . . . . . . . . . . . . 7413.8.1 Idea of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7513.8.2 Exclusion of the sing spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7513.8.3 In- and outgoing spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7513.8.4 Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7613.8.5 Schwartz space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7613.8.6 Compact operators, weak convergence . . . . . . . . . . . . . . . . . . . . . . . . . 7713.8.7 Short range potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7713.8.8 Escape of the wave packet - Perry’s estimate . . . . . . . . . . . . . . . . . . . . . 7813.8.9 Existence of scattering operators for short range potentials . . . . . . . . . . . . . 8013.8.10 Compactness — in- and outgoing parts are almost free wave packets . . . . . . . . 8113.8.11 Final steps of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

13.9 Coulomb scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

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13.9.1 Exact solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8413.9.2 Coulomb + finite range potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

14 Formal derivations 8414.1 The Lippmann-Schwinger equation in disguise . . . . . . . . . . . . . . . . . . . . . . . . . 8414.2 Ω± in the spectral representation of H0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

14.2.1 Remark on “ac-spectral eigenfunctions” . . . . . . . . . . . . . . . . . . . . . . . . 8614.2.2 The LS equation in position space . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

14.3 Mapping free to free spaces: the S-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 8714.3.1 Spectral representation of the S-matrix . . . . . . . . . . . . . . . . . . . . . . . . 8714.3.2 The T -matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

14.4 Scattering cross section — derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8814.4.1 Post and prior form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

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Preliminary remarks

I was remarked that students are so overwhelmed by the formalism of quantum mechanics that they haveno chance to think what it implied. The just have to “shut up and calculate”.

I absolutely agree with that statement. If it were not for practical necessity, one should ban theway that quantum mechanics is (and inevitably must be) taught in a first course. In its epistemologicalcontent it is often communicated by indoctrination rather than scientific reasoning. In bachelor quantummechanics, we struggle all the time in the foothills of a mountain of math. To see what really matters(and what not), we need to climb that mountain.

The root of the problem with quantum mechanics lies here: we talk about reality and of properties ofobjects most of the time in a naıve way and most of the time this is a very reasonable approach. Significantpart of the history of philosophy, however, has been obsessed with the question how to logically capturea concept of “reality” and where to place into that reality what we call “I” (or “we” for that matter) andjust how “real” it is that we see and seem and how such a possible (if possible) real world would makeits way into consciousness.

Physics does not answer these questions. However, in the form of quantum mechanics, it has treadedupon their beginnings in a more striking and inescapable way than ever since the rise of modern sciencewith Galileo and Newton. If there is any belief in physics, then it is that what we know about the worldis not what we can imagine, but what we can reproducibly observe. What we call reality, in physics, isa plausible abstraction from these facts, usually cast into the shape of a logical system, a mathematicaltheory.

In classical mechanics, key elements of reality in that sense - velocity (later more precisely: momentum)and position - appear to have been abstracted incorrectly from observation. In the system of classicalmechanics they both are considered as independently connected to observation. This was always known tobe an idea rather than a thoroughly studied fact. With increasing accuracy and diversity of measurement,we became aware that there is no foundation in experiments for such an assumption. Even worse, by theexperimental violation of Bell’s inequalities, we know that the assumption is wrong.

The first few lectures of this course report a construction of classical and quantum mechanics basedon C∗ algebra that employs many fewer ingredients and assumptions than the usual approach to eithertheory. We cannot claim that there is all facts and no beliefs in this construction. However, this is truefor classical as well as quantum mechanics. We can claim, though, that quantum mechanics results, ifwe omit from classical mechanics one belief that is not at all founded in observation: that we could, inprinciple, do all possible observations on one and the same preparation of a system.

The use of Hilbert space and operators for doing quantum mechanical computations follow from thatvery elementary difference to classical mechanics as a convenient representation of the underlying reality,considering reality as that “plausible abstraction from known facts”.

Once we have reached that point, we will have gone through much of what is needed to deal withbounded operators and we will have a rather complete mathematical foundation for many formal manip-ulations that are commonly used in quantum mechanics.

A decisive next step is to proceed to unbounded operators that represent abstracted, but centralobjects such as position, momentum, or energy. This is rather heavy functional analysis of some mathe-matical beauty and great practical value.

Finally, we deal with scattering theory, which is an important topic but neglected in basic quantummechanics for its technical difficulty. On the one hand, it is esthetically satisfactory to lay a foundationfor the many formal manipulations of scattering theory with its mysterious “scattering states” and eigen-functions of the continuum. In personal experience, I found such knowledge also of immediate practicaluse for analytical work, and even for computational realization.

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1 The C∗-algebra approach to classical and quantum physics

1.1 An analogy

Quantum Classical

State Vector Ψ from H Prob. distr. ρ(x, p) on phase spaceProperty “Observable”, lin. op. A on H Function A(x, p) on phase space

Measur.val. a Eigenvalues of A Function values of A(x, p)Probab. for a

∑|a〉∈Ker(A−a) |〈a|Ψ〉|2

∫Ker(A−a) dx dp ρ(x, p)

After result a Vector Φa ∈ Ker(A− a) Prob. distr. φa(x, p) on Ker(A− a)

Dynamics(I) i ddt

Ψ(t) = HΨ(t) ddtρ = LHρ (Lie-derivative)

Dynamics(II) Ψ(t) = UtΨ(0), 〈Ψ(t)|Ψ(t)〉 ≡ 1 ρ(t) = Φt[ρ(0)],∫dxdp ρ(t) ≡ 1

(I) and (II) are equivalent alternatives

The symbol Φt designates a probability conserving, differentiable map, a flow on phase space. Theparticular Lie-derivative LH may be more familiar as the Poisson bracket.

The formal analogy above is substantially due to an underlying structure that is shared by classicaland quantum theory: observables A are part of a C∗ algebra and states are positive bounded linearfunctionals on that algebra. The only difference between the two theories is that the algebra for classicalmechanics is commutative, but not in quantum mechanics. From this point of view, the Hilbert space isonly a convenient tool to represent (in the sense of representation theory) that structure, in case that thealgebra is non-commuting.

The following discussion relies on:F. Strocchi, Mathematical Structure of Quantum MechanicsW. Thirring, Mathematical Physics III: Quantum MechanicsJ.v. Neumann: Mathematische Grundlagen der Quantenmechanik

1.2 C∗-algebra

A C∗-algebra abstracts the essential algebraic and topological properties of matrices and bounded oper-ators on a Hilbert space:

Definition: AlgebraAn algebra over the complex numbers C:

1. A is a vector space

2. there is an associative multiplication: (P,Q)→ O =: PQ ∈ A with the properties

P(Q1 + Q2) = PQ1 + PQ2 for P,Q1,Q2 ∈ AO(PQ) = (OP)Q

P(αQ) = α(PQ) for α ∈ C∃1 ∈ A : 1Q = Q1 = Q

Definition: ∗-AlgebraAn algebra is a called a ∗-algebra if there is a map ∗ : A → A with the properties (PQ)∗ = Q∗P∗,(P + Q)∗ = P∗ + Q∗, (αQ) = α∗Q∗, Q∗∗ = Q. The element Q∗ is called the adjoint of Q.

8

Obviously this abstracts what you have encountered as “hermitian conjugate” of a matrix or an op-erator.

Definition: C∗-AlgebraAn *-algebra A is called C∗ if it has the following properties:

1. There is a norm 0 ≤ ||Q|| ∈ R with the usual properties (positivity, compatible with scalar multi-plication, and triangular inequality) of a norm on a vector space and in addition

||PQ|| ≤ ||P|| ||Q||||Q∗|| = ||Q||||QQ∗|| = ||Q|| ||Q∗||||1|| = 1.

2. The algebra is complete w.r.t. to that norm (Banach space).

We will discuss these properties later in more detail.

1.3 Classical mechanics

Before discussing this in more abstract terms, let us note that actually the observables of classical me-chanics are part of a C∗ algebra, at least if we restrict ourselves to what can be really measured. Underthe same restriction, states of classical systems are (positive) linear functionals on that algebra.

1.3.1 Observables

The idea of an “observable” is that is represents an actual measurement. In that sense, a classicalobservable is a function on phase space that can be realized in one specific experimental setup. Its valuesA(~r, ~p) is the number extracted from the measurement, given the system is at the phase-space point~r, ~p ∈ Γ, e.g. Γ = R2n The requirement of realization in an experiment restricts observables in this senseto bounded functions:

||A|| := supΓ|A(~r, ~p)| <∞, (1)

which also defines our norm. We see that this operational definition excludes much of what we wouldconsider “properties” of a system (position, momentum, energy, etc.), but it includes everything wecould ever measure in an experiment, which is the more scientific statement. We will re-encounterthis fundamental problem of abstraction, when we will deal with the unbounded operators of quantummechanics.

The rest is easy and it will be left as a first exercise to construct the C∗ algebra that contains theclassical observables.

The abstract “properties” of a system can be considered as limits of the observables. These limitsmay exist, but they will not in general be themselves observables, because the series is not convergent inthe algebra norm. If a sequence of measurement arrangement is convergent, then, by the completenessof the C∗ algebra, the limit is also an observable.

1.3.2 States

As an idealization, the state of a classical systems is fully defined by a point in Γ. However, this is notwhat any scientist ever has been able to observe. If we restrict ourselves to what we can really do, wearrive at a slightly different “operational” definition of a state: It is a procedure that we can repeat manytimes with what we consider high precision. That “preparation procedure” defines what we call a state.

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Repeated observations of the same observable using the same preparation procedure will give a certaindistribution of observed values. We can compute the expectation value of the observables for a givenpreparation procedure ω as the average over many measurement results ai:

〈A〉ω =1

N

N∑i=1

ai ∈ R (2)

The individual value of one measurement is of little significance because of the statistical uncertainty ofour preparation procedure.

As a minimal requirement for “high precision” (for the purpose of a given experiment) is that themeasurement values scatter little, for example that

〈An〉 ≈ 〈A〉n (3)

In that sense, a state is a functional on the observables, more generally on the C∗-algebra that containsthe observables: it maps any element A ∈ A into 〈A〉ω ∈ C, in physics terms: the state (=preparation)of the system determines the average value of a measurement (and also its variance etc.). We usuallyconstruct observables as to be real numbers.

In our specific example we see three properties that we will consider the defining properties of a state,linearity, normalization 〈1〉ω = 1,∀ω, and positivity:

Definition: State of a systemA state is a normalized positive linear functional on the the C∗ algebra of bounded functions.

1.4 Justification of the C∗ algebraic nature of observables

Above we have seen that classical mechanics is “embedded” in a C∗ algebra, i.e. any observable thatcorresponds to realizable experiments can be considered as an element in a larger structure, which forms aC∗ algebra. The assumptions underlying this are very “natural”, in the sense that this is how we perceiveour experimental procedures. However, one must warn already here, the C∗ structure cannot be fullymotivated by such considerations: it may well be that there is a meaningful (maybe even the ultimatelycorrect?) description of nature that cannot be embedded into a C∗ algebra but still fulfills all proceduralrequirements (linearity, normalizability, etc.) that we put a priori into a scientific theory. The Hilbertspace formulation follows from the C∗-structure and the definition of states. It is a convenient technicalrepresentation of the algebra and it is guaranteed to have the essential properties of meaningful theory.There is further structure, though, not required by reason a priori. Therefor this specific form may notbe the only choice. However, we find to this point no statements in the theory that would contradictexperiment.

We will now justify a few aspects of the algebraic form of the physical theory and than make the leapof faith that it is best to realize these properties as part of a C∗ algebra. Much of the justification wasalready anticipated in the discussion of the classical mechanics case.

1.4.1 Observables

An observable A is an experimental apparatus that produces real numbers a as its results. Clearly, wecan, e.g. just by changing measurement units, multiply an observable by a real number λ. We can formpowers of the measurement results by forming the power of the result a. We can define an observable Aas positive if any measurement results in a non-negative number a.

1.4.2 State

A state ω of a physical system is a procedure of repeated experiments that yields a (real) expectationvalue ω(A) for any observable. In the sense of empirical science, a state is completely characterized, if we

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know the expectation values of all possible observables. Two states that give the same result for anythingwe measure are plausibly called equal. By its definition as the average over repeated measurements, wefind the properties for a given observable A:

ω(λA) = λω(A), ω(Am + An) = ω(Am) + ω(An) (4)

1.4.3 Normalization and positivity of a state

The propertiesω(1) = 1 and ω(A) = ω(B2) ≥ 0 (5)

follow easily.

1.4.4 Equality of observables

We can reason in a similar way for the observables: any two observables, that give the same result for allpossible states defined are equal:

ω(A) = ω(B) ∀ω ↔ A = B. (6)

Here we assume that a state can be applied to all observables. This is not completely obvious if weadhere to a strict operational description as a certain way of preparing a systems may be incompatiblewith certain ways of measuring things. Of course, as soon as we abstract the procedure into some internalproperty of the “system” we should be able to perform any possible measurement on it.

1.4.5 C∗-structure

Linear structure and norm can be plausibly deduced along the same lines (see Strocchi): for the linearstructure, we use that we call two observables equal, when their expectation value for all states ω is equal.This, in particular, means that we know an observable, if we know its expectation value for all states

A+B = C : ω(C) = ω(A+B) ∀ω (7)

We can define the norm of an observable as “the largest possible expectation value in any experiment”.As it is not a priori obvious whether this largest value will ever be actually assumed by for a single state,we use the supremum sup rather than the maximum:

||A|| = supωω(A). (8)

Problem 1.1: S how that this has the properties of a norm in a linear space.One may ask, however whether the observable C = A + B can be mapped onto a single realizable

experiment. If not, it may define an object that does not comply with the original definition of observables.Our forming of powers of C relies on this operational prescription: we need to take powers of the individualmeasurement results ai + bi. Now, if we cannot obtain ai and bi for the same individual measurement i,we have no procedure to form a power of C by averaging over powers of ci = ai + bi.

At this point we withdraw from the real world of experimental arrangements into mathematics byassuming that, for whichever are the mathematical objects that represent an observable C, the powers ofC can be formed in a meaningful way. This is certainly the case in classical and quantum mechanics, butwe see no reasons for considering it an a priori necessity of a theory describing experimental practice.

If we accept this, we can make a few more steps to prove the triangular inequality for the norm, butthis is where our beautiful story ends. One can proceed to define symmetric products using (A + B)2 as

(A + B)2 −A2 −B2 = AB + BA =: A B (9)

which results in an algebraic structure that reproduces much of quantum mechanics.

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In some, but not all cases, such an algebraic structure with a symmetric product can be realizedas a subset of a C∗ algebra (embedded in the C∗-algebra). It is no obvious, whether cases where thiscannot be done may be physically interesting. We terminate our discussion here and assume that theobservables are embedded in a C∗-algebra. Our goal is a different one: we will show that the structure ofa C∗-algebra together with a state generates a Hilbert space and that, actually, any Hilbert space witha set of (bounded) observables on it is a (possibly infinite) is equivalent to a sum of such Hilbert spacerepresentations for several states ωi. This includes classical mechanics, which has commutative algebraof observables.

1.4.6 Basic structure of a physical theory

1. A physical system is defined by a C∗-algebra that contains all its observables

2. A state is a positive normalized linear functional on the C∗ algebra

3. An observable is fully defined by its expectation values on all possible states. Conversely, a state isfully defined by its expectation values for all elements of the algebra.

Note:

• There is a duality between states and observables, as we know it from linear functionals on vectorspaces.

• In reality, we know exactly two theories that comply with this definition: classical and quantummechanics. In case of classical mechanics, it appears to be an overblown definition, but it preciselyfits quantum mechanics as we know it.

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2 C∗ algebra

Definition: AlgebraAn algebra over the complex numbers C:

1. A is a vector space

2. there is an associative multiplication: (P,Q)→ O =: PQ ∈ A with the properties

P(Q1 + Q2) = PQ1 + PQ2 for P,Q1,Q2 ∈ AO(PQ) = (OP)Q

P(αQ) = α(PQ) for α ∈ C∃1 ∈ A : 1Q = Q1 = Q

Definition: ∗-AlgebraAn algebra is a called a ∗-algebra if there is a map ∗ : A → A with the properties (PQ)∗ = Q∗P∗,(P + Q)∗ = P∗ + Q∗, (αQ) = α∗Q∗, Q∗∗ = Q. The element Q∗ is called the adjoint of Q.

Obviously this abstracts what you have encountered as “hermitian conjugate” of a matrix or an op-erator.

Definition: C∗-AlgebraA *-algebra A is called C∗ if it has the following properties:

1. There is a norm 0 ≤ ||Q|| ∈ R with the usual properties (positivity, compatible with scalar multi-plication, and triangular inequality) of a norm on a vector space and in addition

||PQ|| ≤ ||P|| ||Q||||Q∗|| = ||Q||||QQ∗|| = ||Q|| ||Q∗||||1|| = 1.

2. The algebra is complete w.r.t. to that norm (Banach space).

2.1 Banach space

Definition: Linear spaceA linear space (≡ vector space) over the complex numbers C is a set where an addition and multiplicationby scalars is defined.

Examples (we will use these to illustrate the distinctive properties of C∗-algebras)

1. Vectors in Cn

2. Complex n× n matrices

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3. Polynomials of complex arguments

4. Cr - the set of r-times continuously differentiable functions

5. Analytic functions

Definition: A normed spaceis a vector space V where a norm || · || : V → R is defined with the properties

1. ||~v|| ≥ 0

2. ||α~v|| = |α| ||~v||

3. ||~u+ ~u|| ≤ ||~u||+ ||~v|| (triangular inequality)

4. ||~v|| = 0⇔ ~v = 0

Examples of norms

For the vector spaces we can define norms in various ways

1. Cn, the p-norms

||~v||p =

[n∑i=1

|vi|p]1/p

, 1 ≤ p ≤ ∞

As a limiting case p =∞:||v||∞ = max

i|vi|

2. On the n× n-matrices M , we also can define (entry-wise) p-norms

||M ||p =

n∑i,j=1

|mij |p1/p

An interesting case is p = 2 (Hilbert-Schmidt norm), where we can write

||M ||2 =

n∑i,j=1

|mij |21/2

=[Tr(MM∗)

]1/2(We denote by ∗ the adjoint = hermitian conjugate). If we were to arrange the entries of the matrix

in a vector, say ~vM , this becomes ||M || = ||~vM ||2.

3. On the n× n-matrices M , we may define the norm

||M || = sup||~v||=1

||M~v||,

where for the vectors ||v|| and ||M~v|| we use the 2-norm. For linear operators, this is the “operatornorm”, in case of matrices the largest magnitude of the eigenvalues.

As the space is finite, we could just as well have written a max instead of sup. For finite matrices,although this matrix norm and all entry-wise p-norms are distinct, they are equivalent in theirtopological consequences: any open set with respect to one norm contains an open set w.r.t. to theother norm. The analogous constructions in infinite Hilbert spaces generate profoundly distinctlimiting behavior and the choice of a “suitable” norm will be essential.

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4. Polynomials P (x) on a compact set, supremum norm.

5. r-times continuously differentiable functions, supremum norm.

6. If a measure for integration is defined: generalization of the p-norms.

Definition: CompletenessA space is called complete if any Cauchy-sequence has its limit in the space. A normed space that iscomplete w.r.t. the norm is called Banach space.

2.1.1 Comment

Here we begin to differentiate a bit more: of the examples above, the finite-dimensional spaces are com-plete (they always are), further the continuous functions on a compact set (r = 0) and the functions withthe p-norm. Polynomials do not always converge to polynomials, similarly one easily looses differentia-bility, r > 0, in a limiting process.

2.2 Classes of elements of a C∗ algebra

1. Hermitian: Q∗ = Q

2. Unitary: Q∗Q = QQ∗ = 1

3. Normal: Q∗Q = QQ∗

4. Projector: Q2 = Q = Q∗ (more strictly: orthogonal projector)

5. Positive: ∃C ∈ A : Q = C∗C

2.3 Open set

If a norm is given on some set X, we can define an open ball of radius r > 0 around z0: Br(x) = z0 ∈X|||z0 − y|| < r. We can then define:

Definition: Open setis U ⊂ X, where for any u ∈ U there exists Bε(u) ⊂ U .

We see that the norm allows to construct to define a system of open sets for X, one says the norm“induces a topology”.

2.4 The spectrum of A ∈ A

Definition: Resolvent set and spectrumThe set

z ∈ C : ∃(A− z)−1 (10)

is called resolvent set of A. Its complement σ(A) is called the Spectrum of A.

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2.4.1 Series expansion of the resolvent

(z −A)−1 = z−1∞∑n=0

(z−1A)n. (11)

That series, depending on the size of |z|, may or may not be convergent.

2.4.2 The resolvent set is an open set

Let z0 be in the resolvent set and let |z − z0| < ε < 1/||(z0 −A)−1||. Then the formal series

(z −A)−1 = (z − z0 + z0 −A)−1 = (z0 −A)−1∞∑n=0

(z − z0)(z0 −A)]−1

n(12)

is norm-convergent and therefore has its limit in the C∗ algebra. To see norm-convergence, we need tothe estimate

||N∑n=0

[(z − z0)(z0 −A)]−1|| ≤︸︷︷︸triangular

N∑n=0

||[(z − z0)(z0 −A)]−n|| (13)

≤︸︷︷︸||PQ||≤||P||||Q||

N∑n=0

|(z − z0)|n ||(z0 −A)]−1||n (14)

and choose |z − z0|||(z0 −A)−1|| < 1 or z ∈ Bε(z0) with ε = 1/||(z0 −A)−1||

2.4.3 Algebra element properties and spectrum

There is a close relation between the spectrum and the class of an element and its spectrum. In fact, thespectral properties fully characterize hermitian, unitary, positive and projector elements, if we assumethat Q is normal. You will study this as an exercise.

2.5 Problems

The first few of the following problems are drawn from the “Preliminaries” section of Reed-Simon, whereyou also can find much of the solutions. It may be (I hope) that you are familiar with some of the math,in that case consider the exercises as reminders.

Problem 2.2: Normed vector space Let C[0,1] denote the continuous real-valued functionson [0, 1].

(a) Convince yourself that with pointwise addition this is a linear space.

(b) Show that the functionals

||f ||∞ = maxx∈[0,1]

|f(x)| and ||f ||1 =

∫ 1

0

dx|f(x)| (15)

are norms.

(c) Generalize the norm and proof of the norm properties for the space continuous functions A(~r, ~p)over a compact subset of phase-space Γ ⊂ R6.

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Problem 2.3: Convergence, Cauchy sequence Remember the definition of convergence:a sequence of elements xn, n = 0, 1, . . . ,∞ from a normed (vector-) space V is said to converge to anelement x ∈ V if ||x − xn|| → 0 as n → ∞. Two other important concepts are open ball Bε(x) =y∣∣ ||y − x|| < ε and neighborhood of x, Nx : ∃ε and Bε(x) ⊂ Nx, i.e. a set that contains x and also

an open ball around x.

(a) Prove that any convergent sequence is a Cauchy sequence.

(b) xn → x if and only if for each neighborhood Nx of x there exists an M such that m ≥ M impliesxm ∈ Nx.

Problem 2.4: Operator norm Let V denote a vector space over the complex numbers C witha norm || · ||. Let T denote a linear transformation V → V

T : Tv = u, (16)

(a) Show that linear transformations form themselves a linear space B, if one defines addition of twomaps by

T, S ∈ B : (T + S)v = Tu+ Su (17)

and in similar way multiplication by a scalar.

(b) Assume now that V is finite-dimensional (i.e. there is a finite basis) Show that

||T || := maxv∈V||Tv||/||v|| (18)

defines a norm on B (||Tv|| is the norm on V).

(c) Consider the case where V is infinite-dimensional. Show that

||T || := supv∈V||Tv||/||v|| (19)

can diverge, i.e. not every linear operator is bounded. Give the simplest possible examples forthis with V = L2(dx,R). On l2 (= “infinitely long vectors” (a0, a1, ...an, ...), can you give aninfinite-dimensional matrix that is unbounded?

Continuity is an essential property of maps, as it allows us to take limits, i.e. to approximate, e.g. anidealized measurement by a sequence of increasingly accurate measurements and assume that the perfectmeasurement would be close to our “increasingly accurate” measurements. In a sense, it is the idea of“continuity” makes the concept of “accuracy” meaningful.

For functions f between sets with a norm, continuity can be best understood as “the limit of thefunction values is the function value of the limit”, i.e.

f is continuous ⇔ f(xn)→ f(x) if xn → x. (20)

This is not the most general definition of continuity, but it serves all our purposes and matches intuition.

A bounded operator is an operator with finite operator norm (problem above). In linear spaces,boundedness of a linear map and continuity are intimately related.

Problem 2.5: Boundedness and continuity Let E and F be two normed linear spaces and

let B : E→ F be a linear map. Show that the following statements are equivalent:

1. B is continuous at the point 0 = x ∈ E

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2. B is bounded

3. B is continuous everywhere

w.r.t. to the supremum-norm.

Hint: This is a standard result for linear maps. For those who have not been exposed to it, here a fewhints: show 1⇒ 2⇒ 3⇒ 1. In the first step use the fact that in a linear space, any vector ||x|| = 1 can

be scaled to length ε. Apply this to ||Bx||/||x|| that appear in the definition of the norm. For the stepbounded⇒ continuous, observe that ||B(xn−x)|| < ||B||||xn−n|| by construction of the operator norm.

Problem 2.6: Continuity of maps, completeness Let E and F be two Banach spaces andlet B(E,F) be the space of the continuous linear maps from E to F.

(a) From the previous exercises we know that the linear maps themselves form a linear space. Show

that ||B||B := supx∈E ||Bx||F/||x||E, B ∈ B is a norm for B. Here || · ||E, || · ||F denote the norms inthe respective spaces.

(b) Show that B with || · ||B is a Banach space.

Hint: For completeness of B one needs to show that a Cauchy sequence Bn defines a continuous

linear map B := limn→∞ Bn. Using completeness of F, one sees that it definitely defines a mapinto F. Boundedness can be shown with a little effort, which then also implies continuity by theprevious problem.

Problem 2.7: Positivity Show that for hermitian A the element ||A||1±A is positive .Hint: Write ||A|| ± A = C2 for hermitian C and explicitly construct C using the fact that there is aconvergent Taylor series for

√1± x. Use the properties of the algebra to show the series converges with

a limit in the algebra.

Problem 2.8: Positivity and boundedness Show that any positive linear functional on aC∗ algebra is bounded.Hint: Use Cauchy-Schwartz and positivity of ||A|| ±A for hermitian A.

Problem 2.9: Resolvent Let A ∈ A be an element of a C∗ algebra. Then also A− z ∈ A, wherewe use the notation A−z for A−z1, z ∈ C. We call (A−z)−1 : (A−z)−1(A−z) = (A−z)(A−z)−1 = 1the resolvent RA(z) of A at z if it exists and (A− z)−1 ∈ A.

(a) Show that the formal series expansion

(z −A)−1 = z−1∞∑n

(Az−1)n (21)

converges for |z| > ||A||.Hint: Use the properties of the norm on a C∗-algebra to show the the series of finite sums isCauchy.

(b) Conclude from the above that (A− z)−1 ∈ A for ||A|| < |z|.

Problem 2.10: Classes of algebra elements and spectrum Normality is an importantproperty that is not at all guaranteed.

(a) Using the C∗ algebra of n× n matrices, give an example (the example ;-)) of a non-normal matrix.

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Properties like hermiticity, projector, unitarity, positivity of elements of a C∗ algebra are reflected in thespectra of the respective elements. This is trivial once we can use the spectral theorem, but it can beproven directly as well. For the following cases this can be done without any particular tricks:

(b) σ(A∗) = σ(A)∗

Hint: Of course, you should prove this for the complement of the spectrum, the resolvent set, first.

(c) σ(U) ⊂ (unit circle) for U unitary.Hint: Determine ||U || and ||U−1|| and Taylor-expand (U − z)−1 and (U∗ − z∗)−1 = (U−1 − z∗).

Problem 2.11: Trace For operators in infinite dimensional Hilbert space, the trace formally definedas in finite dimensions

Tr A =

∞∑i=1

〈i|A|i〉, (22)

for any orthonormal (ON) basis |i〉. Verify formally the following properties:

(a) The definition is independent of the choice of the basis |i〉.

(b) Tr AB = Tr BA and Tr ABC = Tr BCA.

(c) Does Tr ABC = Tr BAC hold?

(d) For normal operators in finite dimensions det[exp(A)] = exp Tr A.

New issues arise in infinite dimension:

(e) Is the trace guaranteed to exist? At which places does one need to treat the fact that the formalsums are infinite-dimensional?

(f) Suppose |Tr A| <∞ and |Tr B| <∞, does |Tr AB| <∞ hold?

(g) Suppose |Tr A| <∞ but |Tr B| =∞, can Tr AB exist? (Try to find an example).

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2.6 State on a C∗ algebra

A linear functional ω on a C∗ algebra A is called positive if ω(A∗A) ≥ 0∀A ∈ A. If ω(1) = 1, we call ita state.

Examples

1. In quantum mechanics: 〈x|Ax〉

2. In classical mechanics:∫

Γdxdpρ(x, p)A(x, p) for ρ ≥ 0.

3. On matrices, Tr ρA for a “density matrix” ρ, i.e. a positive matrix (equivalent hermitian withnon-negative eigenvalues) with Tr ρ = 1.

4. Positive measures on function spaces

2.6.1 Notes

1. There holds a Cauchy-Schwartz inequality, i.e.

|ω(A∗B)|2 ≤ ω(A∗A)ω(B∗B) (23)

This will be left to you as an exercise.

2. Positivity of ω implies its boundedness.

3. As always with linear maps, boundedness implies continuity. Intuitively, if two vectors do not differtoo much, ||P −Q|| < ε, the also the images do not differ too much: ω(P −Q) ≤ Cω||P −Q||, whichis the motivation for general definition of continuity: the pre-image of any open set is an open set.(See exercises).

Problem 2.12: Cauchy-Schwartz Prove the generalized Cauchy-Schwartz inequality for statesω

|ω(A∗B)|2 ≤ ω(A∗A)ω(B∗B) (24)

Hint: This can be modeled after one standard form of proving Cauchy-Schwartz. Follow these steps:(a) study 0 ≤ ω((A∗ + B∗)(A + B)), (b) show that ω(A∗B) can be considered as real without loss ofgenerality; (c) use positivity of ω to show that ω(A∗B) = ω(B∗A), (d) look at the square of (a).

2.6.2 The states form a convex set

ω = αω1 + (1− α)ω2, α ∈ [0, 1], ω1, ω2 states (25)

is a state. They do not form a linear space, e.g. 2ω is not a state, if ω is one.

2.6.3 Pure and mixed states

There are special states, the pure states, which cannot be written as a linear combination of states.Conversely, one can prove that any state can be written as a convex combination of pure states:

ω =∑i

αi~ui,∑i

αi = 1, νi pure (26)

States that are not pure are called mixed.

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We know and we will provide the prove later that a quantum mechanical wave function ψ defines apure state ωψ

ωψ(A) = 〈ψ|A|ψ〉.

We see that this, according to our general principles, is not the only possibility for a state. The moregeneral mixed states come by the name of density matrix that is

ωρ =∑i

ρiωψi ,∑i

ρi = 1, (27)

with the actionωρ(A) =

∑i

ρiωψi(A) =∑i

〈ψi|A|ψi〉 (28)

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3 Gelfand isomorphism

The essence of this isomorphism is that Abelian C∗ algebras are isomorphic to algebras of functions andfurther that the algebra spanned by 1, A,A∗ for normal A is equivalent to the functions over σ(A).This contains the essence of the spectral theorem for normal bounded operators.

Definition: Abelian algebrais an algebra where the product is commutative.

In general, in mathematics, the term “homomorphism” designates a map from algebra into anotherthat preserves the algebraic operations in the sense that the image of a sum is the sum of the images,the image of a product is the product of the images. For *-algebras, one defines the

Definition: Algebraic *-HomomorphismA map π : A → B between *-algebras A,B is called *-homomorphism if

1. π(AB) = π(A)π(B)

2. π(αA+ βB) = απ(A) + βπ(B)

3. π(A∗) = π(A)∗.

Definition: CharacterAn algebraic *-homomorphism χ of an Abelian C∗ algebra into the complex numbers C is called char-acter. We denote the set of all characters of an algebra A as X(A).

Illustration

1. Let ~e be a joint eigenvector of all elements the algebra formed by commutating matrices. Thenχ~e(A) = ~e · A~e is a character of the algebra.

2. Bounded functions on a compact set: values at a given element from the compact set.

Comments

1. (χ(A)− z)−1 exists if z 6∈ σ(A), therefore σ(χ(A)) ⊂ σ(A).

2. A character is trivially positive χ(A∗A) = χ(A)∗χ(A) ≥ 0. Therefore it is also bounded.

3. Any character is a state: it is a positive, linear map A → C and χ(A1) = χ(1)χ(A) showsnormalization.

4. Just as the states do not form a linear space (why?), the characters do not form a linear space.

5. As states, also characters are bounded by ||A||: |χ(A)| ≤ ||A||.

6. The Cauchy-Schwartz inequality that holds for states and therefore for characters |χ(A∗B)|2 ≤χ(A∗A)χ(B∗B) implies |χ(A)| ≤ ||A||: choose B = 1 and use that 0 ≤ χ(||A||2 − A∗A) = ||A||2 −χ(A)χ(A)∗.

7. Characters are pure states: let χ be a character and ω1 and ω2 two general states, then (α1ω1+α2ω2)(α1 + α2 = 1) will not be a homomorphism

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8. There exists a pure state such that χ(A) = ||A||: this conforms with our analogy that characters

are correspond to eigenvectors and norm of a matrix ||A|| is the modulus of its largest eigenvalue.

One can explicitly construct that state as follows: Given any A ∈ A, we construct the subspaceof all vectors α1 + βA∗A. On that subspace, χs(α1 + βA∗A) = α + βA∗A is a positive functionalwith χs(1) = 1 and χs(A

∗A) = ||A||2. This χs is only defined on the subspace. However (theoremsof Hahn-Banach and Krein) there are states on all of A that behave as χs on the subspace. Nowthat we know that such states exist, let us denote the states with ω(A∗A) = ||A||2 by Z: Z is aclearly a convex set. The extremal states χe ∈ Z are pure states: because of ω(A∗A) ≤ ||A||2,χs = αω1 + (1− α)ω2 implies ωi(A

∗A) = ||A||2, i.e. ωi ∈ Z and χe would not be extremal in Z.

We had not proven that states are bounded. For characters that prove is really easy and all that weneed for now: For hermitian A we know 0 ≤ ||A|| ±A, therefore 0 < ||A|| ± χ(A), i.e. |χ(A)| ≤ ||A||. Forgeneral A, we note that A∗A is hermitian and therefore (using properties of || · ||

||A||2 = ||A∗A|| ≥ χ(A∗A) = χ(A)∗χ(A) = |χ(A)|2. (29)

3.1 Weak *-topology

Strictly a topology on any set is given, if we define all open subsets of the set.Topology is required to define continuity: a function is continuous at a point, if the pre-image (“Ur-

bild”) of any open neighborhood of a image of the point is an open neighborhood. This is an abstractionand generalization of our intuition for continuous functions on Cn.

We will want to define continuous functions from the character set X(A) into the real numbers. Weneed a topology, i.e. we need to define the “open sets” ⊂ X(A). Now the characters are subsets of thelinear functionals on the algebra: they are subsets of the dual space of the algebra.

The open sets in the dual space are defined at those where any point space W ∗ in the set contains aset Bv,ε(W

∗) = U∗ ∈ V||U∗(v)−W ∗(v)| < ε for all v ∈ V .We define when we call a sequence in the dual space as convergent in the sense of this topology: a

sequence of elements χn from the dual space of some vector space V is said to converge in the sense ofweak convergence, if χn(v) converges as a sequence in C for each v ∈ V.

Illustration and comments on weak convergence In finite dimensions most topologies are equiv-alent and also the distinction between the a linear space and the linear functionals on that space is notessential. In infinite dimensions the distinctions become crucial.

1. Continuous functions on a compact set S: a character χx maps a function f ∈ C0 : χ(f) = f(x)into its value at one point. A sequence of characters χxn converges, if the function values f(xn)converge at that point, which, as f is continuous, is the case if xn converge in the sense of thetopology of S. (You may recognize the δ-functions as characters of that algebra.)

2. n-component complex vectors ∈ Cn: we do not have an algebraic structure and therefor no charac-ters. We have a dual, of course. Choosing a basis ei ∈ Cn, any linear functional W ∗ can be writtenas W ∗(v) =

∑iW∗(ei)vi =

∑i wivi, where the wi are components in the dual space. A sequence

W ∗n converges, if each of its components w(n)i converges. This dual is the isomorphic to (“for all

practical purposes the same as”) the original space.

3. On a finite dimensional vector space we can use an alternative concept with equal result: As wehave a norm for the v ∈ Cn and as we can identify every W ∗ ∈ dual(Cn) with some w ∈ C2, wecan use that norm also for the W ∗: ||W ∗|| := ||w||. Sequences that are convergent w.r.t. the norm(“strongly convergent”) are also weakly convergent. In finite dimensions, also the converse is true.

4. In infinity dimensions weak convergence does no longer imply strong convergence. Hilbert space:weak convergence does not imply strong convergence: trivial, but also typical example is ~vn : (~vn)i =δin

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5. Example from physics: a (purely continuous) spreading wave packet converges → 0 in the weaktopology!

A ∈ A is a continuous function on X(A): We are seemingly going in circles: we just introduced thecharacters χ as functionals on A. We can turn the roles around and consider the elements of the algebraas functions on the set of characters

A 3 A : X → C : χ→ χ(A). (30)

As a direct consequence of the construction of the weak-* topology, A is a continuous function w.r.t. the*-topology on X(A).

Problem 3.13: Continuity of A : X(A)→ C Show that the map A : χ→ χ(A) is continuousw.r.t. the weak *-topology on X.

3.2 Gelfand isomorphism

An Abelian C∗ algebra is isomorphic to the continuous functions C(X) on the characters set X = X(A)if we use the weak *-topology for X and the norm C(X) 3 f : ||f || = supχ |f(χ)|.

Isomorphic By isomorphic we denote maps that are bijections and that conserve the essential structuralcharacteristics such as algebraic properties, *-map, and norm. In colloquial language, things that areisomorphic “are the same”, just represented in different ways.

Outline of the proof Even if this remains very incomplete, the reasoning with approximations byalgebras, dense sets, and compact sets if of general value and therefore useful to look at:

1. We have seen that any element A of the algebra defines a continuous function fA ∈ C(X). LetfA, A ∈ A denote the set of these functions. Clearly, as characters are *-homomorphisms, anyalgebraic combination of these fA’s matches the algebraic combination of the corresponding A’se.g.

fAB(χ) = χ(AB) = χ(A)χ(B) = fA(χ)fB(χ), (31)

and similarly for all other algebraic operations. That means that each element of the algebrare-appears as a function on X(A) and that these functions form a *-algebra.

2. Next we convince ourselves that the C∗-algebra norm maps onto the (supremum) norm of fA ∈C(X):

||A|| != sup

χ|fA(χ)| = sup

χ|χ(A)|. (32)

As also all χ are states, |χ(A)| ≤ ||A||. Here we use the fact that all characters are states and,without proof, that the pure state with |ω(A)| = ||A|| is a character, i.e. the supremum over all χis indeed the norm.

3. We have shown in an exercise that fA is a continuous function, i.e. the map A → fA is injectiveinto C(X(A)).

4. The main mathematical task is to show that A → fA is also surjective, i.e. that indeed all con-tinuous functions f ∈ C(X) have a correspondence in the algebra. Rather then proving anythinghere, we appeal to your knowledge of a fact from the theory of ordinary functions (Weierstraß):polynomials approximate any continuous function on a compact set arbitrarily well in the sense ofthe supremum norm: they are dense in the continuous functions. Compactness must be understoodfor the *-topology on X.

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In fact, this holds not only for polynomials but for any algebra of complex-valued functions on acompact set (not proven here). Now our fA definitely form an algebra of complex valued functionson X(A), which is compact. Therefore the fA are dense in C(X). This means that any functionf ∈ C(X) can be approximated by a sequence fAn with An ∈ A. As ||An|| = ||fAn || we know thatAn is a Cauchy sequence and An → A ∈ A with fA = f .

Comment on this sketch: There are a few very important holes in this; but as it is not our mainpurpose to follow up on this, we only list these holes here clearly:

1. The proof that the pure state with ω(A) = ||A|| is a character is missing. This proof will bedelivered later: the GNS construction shows that pure states for abelian algebras indeed produce*-homomorphisms into the complex numbers, i.e. characters.

2. We have not discussed compactness, which for the special case of weak *-topologies can get tricky.Nonetheless we assert that the character set is compact as needed as input for Weierstraß.

3. Even if we take Weierstraß for granted, its generalization for algebras needs to be looked up in mathliterature, e.g. [?]

3.2.1 Terms: Closure and dense sets

Closure of a subset Let D be a subset of a space X where a distance is defined between all elements(metric space). Then the closure D of D is

D = D ∪ x ∈ X|∃dn → x (33)

Dense subset A subset D of a metric space X (space with a norm) is called dense in X if

D = X. (34)

3.2.2 The spectral theorem in nuce

Let A be a normal element of any (also non-commutative) C∗-algebra A and consider the sub-algebragenerated by 1, A,A∗. Normality implies that the algebra of all polynomials of A and A∗ is Abelian.If we consider the closure of the algebra of the polynomials, we obtain again a now Abelian C∗ algebra.(Strictly, one needs to show that commutativity survives the limit). By the Gel’fand theorem, this algebrais isomorphic to the functions on the character set. This implies, in particular, that

∃(A− z)−1 ⇔ ∃(fA(χ)− z)−1∀χ (35)

orσ(A) = ran(fA) (36)

i.e. the “range” of the function fA (=image fA(X)) is just the spectrum of A. Considering now that

fA(χ) = χ(A) ∈ σ(A) (37)

we see that a character maps A into the spectrum. We can also consider polynomials P

fP (A)(χ) = χ(P (A)) = P (χ(A)) == P (χ(A)) = P (a) for a := χ(A) ∈ σ(A) (38)

to see that all f ’s are just functions on the spectrum of A.To make the spectral theorem as you know it complete, we need the Hilbert space. This rabbit we

will magically pull out using just an innocent state for out top-hat.

Problem 3.14: Spectra and Gelfand isomorphism Use Gelfand isomorphism to provethe spectral characterization of hermitian, unitary, positive and projector elements of a C∗ algebra.

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4 Representations in Hilbert space

4.1 Operator norm

For one of the exercises we already introduced the boundedness of a linear map as the property that avector “does not grow too much” when the the map is applied:

||Av|| ≤ C||v|| ∀v (39)

for some C > 0. The smallest possible such number is the norm ||A||:

Definition: Norm of an operatorLet A : H → H be a linear map of a Hilbert space H onto itself. We designate as the norm of A onHilbert space H the number

||A|| = sup||ψ||=1

||Aψ||, (40)

where on the rhs. || · || denotes the vector norm of H. An bounded operator A is an operator with afinite norm ||A|| <∞.

Note: the operator norm has all the properties that we asked from the norm of a C∗-algebra.

4.2 Representation of an algebra

We have dealt with algebras reduced to their abstract properties, i.e. by defining the results of addition,multiplication. In practice, such an algebra is realized by objects like matrices, differential operators etc.One and the same algebra can be realized in different ways. Also, the realization may only represent partof the complete properties (it may not be an isomorphism).

Definition: RepresentationA representation π of a C∗-algebra A is a *-homomorphism from A into the bounded operators B(H) ona Hilbert space H. If ker(π) = 0 ∈ A the representation is faithful. Two representations π1 on H1 andπ2 on H)2 are equivalent, if there is an isomorphism U : H1 → H2 such that

π2(A) = Uπ1(A)U−1 ∀A ∈ A. (41)

4.2.1 Note

• Continuity of π follows from positivity: π(A∗A) = π(A)∗π(A) is a positive operator on H. Thisimplies in particular

||π(A)∗π(A)ψ|| ≤ ||π(A)||2||v||∀||v|| (42)

0 ≤ A∗A ≤ ||A∗A||1⇒ ||π(A)∗π(A)|| ≤ ||(||A∗A||)1|| = ||A∗A|| (43)

0 ≤ (44)

• As representations may not be faithful, we can have ker(π) 6= 0. In that case

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4.3 Irreducible representation

Definition: Invariant subspaceLet V ⊆ H be a subspace of H, i.e. V is a linear space. V is called an invariant subspace of H for arepresentation π, if

~v ∈ V ⇒ π(A)~v ∈ V, ∀A ∈ A (45)

Definition: Cyclic vectorA vector ~c ∈ H is called cyclic vector for a representation π, if

C := π(A)~c|A ∈ A (46)

is dense in H, i.e.C = H. (47)

Definition: Irreducible representationThe following two properties are equivalent

1. The only closed invariant subspaces V ⊆ H are 0 and H.

2. Any vector φ ∈ H is cyclic.

The first version can als be fromulated as “there are no non-trivial projectors ∈ H that commute withπ(A)∀A ∈ A.

4.4 Sum and tensor product of representations

The concepts of direct sum and tensor product can be introduced more abstractly than what we do here.We, in both cases, assume that our Hilbert spaces are equipped with a denumerable bases |i〉 (it is aseparable Hilbert space) and use these for the following definition.

Definition: Direct sum of Hilbert spacesLet J and K be Hilbert spaces with the bases |j〉 and |k〉. Then the space spanned by the basis

|i〉⊕ = |ji〉 ⊕ |ki〉 (48)

with the addition rules

|m〉⊕ + |n〉⊕ = |jm〉 ⊕ |km〉+ |jn〉 ⊕ |kn〉 = |jm + jn〉 ⊕ |km + kn〉 (49)

and the scalar product〈m|n〉⊕ = 〈jm|jn〉+ 〈km|kn〉 (50)

and analogously for the multiplication by a scalar α ∈ C is a Hilbert space. Show completeness

Definition: Tensor product of Hilbert spacesLet J and K be Hilbert spaces with the bases |j〉 and |k〉. Then the space spanned by the basis

|i〉⊗ = |ji〉 ⊗ |ki〉 (51)

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and the scalar product〈m|n〉⊕ = 〈jm|jn〉〈km|kn〉 (52)

and analogously for the multiplication by a scalar α ∈ C is a pre-Hilbert space. Its completion

span(|j〉 ⊗ |k〉) = J ⊗K = H (53)

and is called the tensor product of J with K.

4.4.1 Illustration

Definition: Direct sum of operatorsLet B ∈ B(J ) and C ∈ B(K) be operators on two Hilbert spaces. Their direct sum is defined by itsaction on the basis functions:

B(J ⊕K) 3 C|i〉 =: (B⊕C)(|ji〉 ⊕ |ji〉) = (B|ji〉)⊕ (C|ki〉). (54)

As the operators are bounded (=continuous), this can be extended to a definition on the complete space.

Definition: Tensor product of operatorsLet B ∈ B(J ) and C ∈ B(K) be operators on two Hilbert spaces. Their tensor product is defined byits action on the basis functions:

B(J ⊗K) 3 C|i〉 =: (B⊗C)(|ji〉 ⊗ |ji〉) = (B|ji〉)⊗ (C|ki〉). (55)

As the operators are bounded (=continuous), this can be extended to a definition on the complete space.

Definition: Direct sum of representationsLet π1 and π2 be two representations of the same algebra on the Hilbert spaces H1 and H2. The directsum of the representations is defined by

π1 ⊕ π2 = π : A → B(H1 ⊕H2) : π(A) = π1(A)⊕ π2(A) (56)

Direct sums of representations are the prototypical form a redicible representations.

4.5 Illustration

• Draw vectors and matrices

• Point out the important differences!

• Remind of tensor product; as to direct sum: archetypical form of reducible representation

4.6 Problems

Problem 4.15: Irreducibility Let π be a representation of an algebra A on a Hilbert space H.Show that the two definitions of irreducibility are equivalent:

1. The only closed invariant subspaces under π are H and 0.

2. Any non-zero vector of H is cyclic.

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Problem 4.16: Positivity and boundedness

(a) Use the Gel’fand isomorphism to argue that A∗A ≤ ||A∗A||1

(b) Convince yourself, that for operators in Hilbert space the definition of positivity translates into〈ψ|Bψ〉 ≥ 0∀ψ ∈ H.

(c) Show that positivity is conserved in representation, i.e. A positive ⇒ π(A) positive

(d) Use the above to show that any representation π : A → B(H) is a continuous map. (Remember therelation between continuous and bounded for linear maps.)

5 The GNS construction

It is easy to see that, given a C∗ algebra (which in particular is a linear space) and a state, one canconstruct a linear space with a hermitian sesquilinear form

〈A|B〉 =: ω(A∗B). (57)

show hermiticity, using positivity. Unfortunately, positive definiteness is missing: 〈A|A〉 = 0 6⇒ A =0. We never excluded that a given ω has 0 expectation value for some observable A∗A. We resolve thisby removing these vectors with zero “ω-length”

√ω(A∗A) = 0.

To make this precise, we need to introduce a few concepts. First, we note that elements of the algebrawith ω-length 0 transmit this property by multiplication:

The left-sided ideal Nω Denote the set of elements with zero length by

Nω = A ∈ A|ω(A∗A) = 0. (58)

Then for any N ∈ Nω we have ω(N∗A∗AN) ≤ ||A∗A||ω(N∗N) = 0, i.e

N ∈ N ⇒ AN ∈ Nω. (59)

(One says: Nω is a left-sided ideal of A).

Proof. • A∗A < ||A∗A||1 - through Gel’fand isomorphism, function is positive,√||A||2 −A∗A =:

C ∈ A exists.

• then 0 < (CB)∗CB = B∗C2B = B∗(||A||2 −A∗A)B or B∗A∗AB < ||A∗A||B∗B.

• by positivity and linearity of states: 0 < ω(||A∗A||B∗B−B∗A∗AB) = ||A∗A||ω(B∗B)−ω(B∗A∗AB)

• N ∈ N , A ∈ A: ω(N∗A∗AN) ≤ ||A∗A||ω(N∗N) = 0

We can now define an algebra, from which N is “divided out” (the quotient algebra): the idea is thatif any two elements differ only by an element of ω-length 0, they are considered as equivalent, both beingthe valid representatives of an equivalence class.

Definition: Quotient spaceWe denote by bB the sets

bB := A ∈ A|∃N ∈ Nω : A = B +N (60)

The set of the equivalence classes b is denoted by A/Nω (the quotient space).

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Equivalence class The relation

A ∼ A′ : A = A′ +N ∈ N (61)

defines an equivalence relation

A ∼ A

A ∼ B ⇔ B ∼ AA ∼ B, B ∼ C ⇒ A ∼ C

The sets bB form equivalence classes, meaning

A ∼ B ⇔ bB = aA (62)

Any representative B from the class is equally good. Note in particular that the 0 ∈ A/Nω correspondsto Nω ⊂ A (it is a true subset, why?).

With this we can construct a pre-Hilbert space

Lemma Let ω be a state, then the quotient space A/Nω with the scalar product

〈bB |bB′〉 = ω(B∗B′) (63)

is a pre-Hilbert space.

Proof. The only non-trivial thing to show is bB = 0 ⇒ 〈bB |bB〉 = 0. Assume bB 6= 0, i.e. B = A + Nwith A 6∈ Nω:

ω(B∗B) = ω(A∗A) + ω(N∗A)︸ ︷︷ ︸|ω(N∗A)|2≤ω(N∗N)ω(A∗A)

+ω(A∗N) + ω(N∗N)︸ ︷︷ ︸=0

> 0 = ω(A∗A).

We know |ω(A∗N)| ≤ ω(A∗A)ω(N∗N) = 0 and we get A/Nω 3 b 6= 0⇒ 〈b|b〉 6= 0.

From this we can obtain a Hilbert space by including all limits w.r.t. to the ω-length into our space,by operating in the closure A/Nω.

We can immediately define an action of our algebra on the vectors of this space: let b ∈ A/Nω andB ∈ A a representative of b, i.e. b = bB . Then

π(A)|b〉 = |c〉, cAB = C ∈ A|C = AB +N,N ∈ Nω. (64)

Strictly speaking this is only defined for the quotient algebra A/Nω, but by continuity we can extend itto being defined also on its completion A/Nω.

5.1 Reducible and irreducible GNS representations

5.1.1 1 ∈ A/N is cyclic

Trivially so, as any A/N 3 b = bB = π(B)1.

5.1.2 Irrep ⇔ pure state

In the exercises you have shown that the GNS representation corresponding to a pure state is irreducible.We do not prove that also the converse holds: if a GNS representation is irreducible, the correspondingstate is pure.

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5.1.3 Irreps for abelian C∗ algebras

For an abelian algebra, any irreducible representation must be one-dimensional. This intuitive, but notcompletely obvious.

As the GNS representation from a pure state is irrep it is one-dimensional, i.e. it is equivalent to a*-homomorphism into C. That implies that a pure state on an abelian algebra is a character. We hadused this for Gel’fand theorem. Note that for GNS we do not make any reference to Gel’fand, ie. theargument is not circular.

Problem 5.17: Pure states and characters Accepting the fact that any irrep of an AbelianC∗ algebra is one-dimensional, show that pure states on Abelian C∗ algebras are characters.Hint: Evaluate for GNS 〈1|(πω(A)|1〉) and note that in 1-d πω(A)|b〉 ∝ |b〉.

5.1.4 Any representation is the (possibly infinite) sum of irreps

Suppose we have a representation in a separable Hilbert space H. Remember that any vector |i〉 ∈ Hdefines a pure state. According to the previous statement, the selected state defines an irrep GNS, andthe GNS space is isomorphic to an invariant subspace of Hi ⊂ H. We can now pick a state |j〉 6∈ Hi andrepeat the procedure to obtain an new irrep on Hj ∩ Hi = 0 (being irreps, if they share one vectorthey need to be identical). As the Hilbert space is separable, the iteration of the procedure exhausts thecomplete space.

Each of the irreps is isomorphic to the corresponding irrep GNS.

Problem 5.18: GNS representation

(a) Find at least one cyclic vector for the GNS representation

Show that a pure state ω produces a irreducible GNS representation πω. We denote by G the GNS space,by B(G) the set of all bounded operators on it, and by |Ω〉 ∈ G the vector that corresponds to 1 ∈ A.Follow the steps:

(b) Show that ω pure ⇔ any positive linear functional µ with µ ≤ ω is µ = λω.Hint: ⇒: convince yourself that ω = µ+ (ω − µ) is a convex linear combination of two statesHint: ⇐: construct µ 6= λω, 0 < µ < ω from ω = αω1 + (1− α)ω2. E.g., imagine a density matrixρ with two entries on the diagonal and construct a matrix that is not proportional to that one, butstill smaller.

(c) Let P 2 = P, P = P ∗ ∈ B(G) be a projector onto an invariant subspace. Convince yourself thatPπ(A) = π(A)P ∀A.

(d) Show that µ(A) : A → 〈PΩ|π(A)PΩ〉 is a positive linear functional on A with the propertyµ(C∗C) ≤ ω(C∗C), ∀C ∈ A, i.e. µ ≤ ω.

(e) Consider the matrix elements 〈π(A)Ω|Pπ(B)Ω〉 and determine the possible values of λ of µ = λω.Conclude that P = 1 or = 0.

Interestingly, also the converse is true, i.e. a GNS representation for a state ω is irreducible if and onlyif ω is pure.

Problem 5.19: Convergence of operators Let B(H) denote the space of bounded operatorson a finite-dimensional Hilbert space H and let Bn denote a sequence of operators. Show that thethree following definitions of convergence are equivalent

1. Norm-convergence: Bn → B ⇔ ||Bn −B|| → 0 (Remember: ||B|| = sup||ψ||=1 ||Bψ||)

2. “Strong” convergence: Bns→ B ⇔ ||Bnψ −Bψ|| → 0∀ψ ∈ H

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3. “Weak” convergence: Bn B ⇔ 〈φ|(Bn −B)ψ〉 → 0∀φ, ψ ∈ H(Note the notation by ).

On infinite-dimensional Hilbert spaces this is not the case, there is only the implication norm-convergence⇒ strong convergence ⇒ weak convergence.

(a) Using H = l2, i.e. the space of infinite length vectors, construct Bns→ B, but Bn 6→ B

(b) Construct Bn B, but Bn 6s→ B.

Hint: In both cases the trick is to let escape the maps Bn to ever new directions, e.g. by a sequence ofoperators that connect ever new pairs of entries etc.

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6 Towards the spectral theorem for bounded operators

We will next derive the spectral theorem for normal operators in Hilbert space. The steps for thisreasoning are as follows:

1. By Gelfand isomorphism, the algebra spanned by the normal operator A is isomorphic to thealgebra of bounded on functions σ(A). In particular, the operator A corresponds to the functionfA(a) = a.

2. We will show that any state ωi defines an integration measure µi(a) on σ(A) by identifying thefunctionals ω with the integral

ω(P (A)) =

∫σ(A)

dµi(a)fP (A)(a)

(By its boundedness, the ω extends to the complete algebra of bounded functions, not only thepolynomials, B.L.T. theorem to be proven).

3. The GNS Hilbert space Hi is then isomorphic to the Hilbert space L2(dµi, σ(A)).

4. As the Hilbert space H of any representation is isomorphic to a direct sum of GNS-representationsHi for different states ωi, it is also isomorphic to a direct sum of L2(dµi, σ(A))

The last statement is what we know as the spectral theorem: there is a invertible, norm-conserving mapfrom any Hilbert space H where a (bounded) operator is defined to a direct sum of Hilbert spaces Hi,where the operator A appears as multiplication by the spectral value. A given subset τ ⊂ σ(A) of thespectrum may appear with measure µi(τ) > 0 in several of the representations Hi: this is what you firstencounter by the name of a “degenerate” eigenvalue.

7 Elements of measure theory

(A representation of this can be found in Teschl: Mathematical Methods in Quantum Mechanic).Loosely speaking a measure on a set X is a function that assigns to any reasonable subset of X a

real number in a way that is additive, i.e. the measures of disjoint sets just add up. “Reasonable” needsto be qualified in the more precise concept of a Σ-algebra. Once this is given, one can use the concept ofthe Lebesgue integral to define integrals over the set.

7.1 Σ-algebra

Definition 1. A Σ-algebra is a set A of subsets of a given set X such that

1. X ∈ A

2. A is closed under countable many unions and intersections, and under complements.

7.2 Borel sets

For a given topology on a space X (i.e. a definition of the open sets, e.g. open balls on a metric space),one can construct a Σ-algebra by just accepting all subsets of X that can result from countably manyset operations. This algebra is called the Borel Σ-algebra and its elements are the Borel sets.

It is useful to consider the following examples of which sets are Borel-sets for the real axis (with thestandard topology)

1. Closed sets

2. Single points as a special case of a closed set or also as⋂∞n=1(x− 1

n , x+ 1n )

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3. The Cantor set is obtained by the following procedure: From the interval C0 = [0, 1], remove themiddle third, i.e. C1 = [0, 1/3] ∪ [2/3, 1]. Iterate by applying the analogous procedure to the thetwo subintervals, take it to infinity. The resulting set has Lebesgue-measure zero and it has anon-countable number of points.

Problem 7.20: C onstruct the Cantor set and show the above properties.Problem 7.21: S how that the Cantor set is Borel and that a function on the cantor set is continuousalthough its derivative = 0 everywhere

Problem 7.22: Cantor set The Cantor set is obtained by recursively removing the open invervals(1/3, 2/3) from the compact interval [0, 1], then again the center third from [0, 1/3] and [2/3, 1] etc.

(a) Show that C is a Borel set with Lebesgue measure λ(C) = 0.Hint: Study its complement

(b) Show that C = limn→∞ Cn, where Cn is constructed by recursion

Cn+1 =1

3Cn ∪ [

2

3+

1

3Cn]

(c) Show that the Cantor set is not countableHint: Write the points in the Cantor set as ternary (base 3) numbers, show that it contains allnumbers with digits 0 and 2).

(d) Show that the Cantor set does not contain any isolated points, i.e. for any point x in C and anyε > 0 there is y ∈ C, such that |x− y| < ε.

7.3 Measure and measurable space

Definition 2. Let Σ denote a Σ-algebra of X. Then a (positive) measure µ is a map Σ → R with theproperty

1. µ(E) ≥ 0 ∀E ∈ Σ

2. µ(∅) = 0

3. For countably many Ei that are mutually disjoint Ei ∩ Ej = ∅, it holds

µ(⋃i

Ei) =∑i

µ(Ei)

The pair (X,Σ) is called a measurable space, Σ are the measurable sets. If a concrete measure isdefined we call the triplet (X,Σ, µ) a measure space.

7.4 A few definitions for the Lebesgue integral (reminder)

Measurable function Let (X,Σ) be a measurable space. A function f : X → R is measurable ifthe pre-image (inverse image) x ∈ X|f(x) > t ∈ R is in Σ. As we are for now mostly dealing withcontinuous functions, this is trivially the case.

Simple function is a finite linear combination of characteristic functions χMkof measurable sets Mk

s =∑k

akχMk(65)

A little caution is required if terms can be infinite. However, we will be dealing with finite measuresµ(X) <∞, where this cannot happen.

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Integral of a positive simple function∫ ∑k

akχMk=∑k

akµ(Mk) (66)

Integral of a positive measurable function∫X

fdµ := sups<f

[∫sdµ, s simple

](67)

Integrals of non-positive and complex-valued functions are obtained by taking the functionapart into positive and negative parts f = f+ + f− and treating the two parts separately. Similarly,integrals of complex valued functions are treated by integrating real and imaginary parts separately.

7.5 Lebesgue measure

A particular form measure on the Borel sets of the real axis is the Lebesgue measure used in standardintegration theory. There, we assign to an (open) interval its length as its measure. This carries over toall intervals and in fact to all sets that can be written as unions of intervals by the additivity. On Borelsets that procedure is guaranteed to work (no proof given here).

On this basis, we can proof

1. Any countable union of points has Lebesgue measure 0.

2. The Cantor set, although not countable, still has measure 0.

7.6 Absolutely continuous

A characteristic of physics is the existence of discrete and (absolutely) continuous spectrum of an observ-able. The typical physical example of absolutely continous spectrum is the spectrum of kinetic energyor any unbound part of the spectrum. We were instructed that wavepackets of the free motion decayby “wave packet spreading”, eventually to arbitrarily far distances. This is in fact a general propertyof packes on the ac spectrum, which we will prove later. In contrast, wavepackets on the discrete spec-trum remain confined, possibly undergoing “revivals”. This close association of basis characteristics withphyiscal gives the distinction of point and ac spectrum its physical relevance.

Mathematically, there can be more than these two kinds, let us call the rest the “singular” spectrum.In fact, such a spectrum may even appear in physical system such infinitely extended, non-periodicsystems, such as created by Penrose tiling of a plane. (See, e.g. a review by Yaron Last on Barry Simon’scontributions).

The notions can be made precise definition and the subsequent theorem.

Definition 3. A measure µ is called absolutely continuous (a.c.) w.r.t. the Lebesgue measure λ onthe real axis, if λ(A) = 0⇒ µ(A) = 0.

7.6.1 Examples

1. An easy example for an absolutely continuous measure is what we usually do when we “change theintegration coordinate”:

dµ(x) =dµ

dxdx,

where dµ/dx is a non-negative function of x. More closely to our present notation we should write

µ(E) =

∫E

fdλ (68)

In fact, this is already the most general form of an absolutely continuous measure.

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2. Clearly, point measures are not a.c.

3. Interestingly, also any measure that is non-zero on a Cantor set is not a.c.: it is neither a.c., nordiscrete (countably many points), it is the “other” mentioned above.

Two different measures µ, ν can be mutually singular. This means that one can find disjoint setsM ∩M = ∅, M ∪N = X such that

µ(N) = ν(M) = 0, (69)

which also implies µ(M) = ν(N) = 1. Loosely speaking: the two measures do not share any subsetswhere both are non-zero.

Now we can formulate

Theorem 1. Lebesgue’s decomposition theorem. Any Borel measure ν on the real line can be decomposedinto

ν = νac + νp + νsing (70)

with the three parts

νac absolutely continuous

νp discrete

νsing singular,

which are mutually singular and νp and νsing are both singular w.r.t. the Lebesgue measure.

7.7 Signed measures and complex valued measures

One can abandon the positivity requirement on the measure and rather admit any real or even complexnumber to be the measure of Borel sets. The most obvious example is to integrate with a signed or evencomplex “weight” function. The property that should be maintained is the additivity. The measure canbe taken apart into imaginary and real parts, finally into the positive and negative parts. For each ofthese at most 4 parts, integration can be defined by the Lebesgue procedure.

7.8 Projector valued measures

One can make another turn of the screw and assign to the measurable sets E (self-adjoint) projectorsP (E) onto some subspace of a Hilbert space H. It should have the properties

1. π(X) = 1, i.e. the measure of the complete measure space is the identity on the Hilbert space.

2. For any two functions φ, χ ∈ H, the map E → 〈φ|P (E)χ〉 should be a complex valued measure.

Problem 7.23: I t follows from the above that the projection values measure of two disjoint sets

E ∩ F = ∅ (71)

are orthogonalP (E)P (F ) = 0 (72)

and more that the measures of any two sets commute

P (E)P (F ) = P (E ∩ F ) = P (F )P (E). (73)

We will use such measures to formulate the spectral theorem more precisely.Integrals using projection values measures have as their results operators.

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7.9 Positive operator valued measures

The idea can be carried even further, although this will not be relevant for us: instead of projectors,on can assign positive operators to the Borel sets such that they add up over all X to unity, but theoperators on disjoint sets do not need to be orthogonal.

7.10 Measures and continuous functionals

It turns out that positive linear functionals on continuous functions (at least on compact sets) all havethe form of an integral for the Borel Σ-algebra.

Theorem 2. (Riesz-Markov) Let X be a compact Hausdorff space. For any positive linear functional ωon C(X) there is a unique (finite) Borel measure µ on X with

l(f) =

∫dµf (74)

Hausdorff space (separated space): “different points are separated”, any two distinct elements fromthe space have disjoint neighborhoods.

Finite measure: the measure of the complete space is finite µ(X) <∞.Of course, the measure is not usually the Lebesgue measure.We see where this is going: our states define a measure on the character set, which is — we have used

it before without proof — a compact set and a separated space.

Problem 7.24: Projection valued measures Let P be a projection valued measure.

(a) Show that for any two Borel sets E ∩ F = ∅ ⇒ P (E)P (F ) = 0.Hint: Use additivity of the measure, the projector property P (E)2 = P (E) and study [P (E) +P (F )]2.

(b) From that, conclude that the measures of any two sets commutes P (E)P (F ) = P (F )P (E).Hint: Split the sets into E = E′ ∪ S, S = E ∩ F etc.

Problem 7.25: Operators as integrals with projection valued measures

(a) Show, formally (i.e. without considering the existence of a limit), that any hermitian boundedoperator on a Hilbert space H can be written in the form of an integral with a projection valuedmeasure on its spectrum

A =

∫σ(A)

αdP (α) (75)

Hint: Construct the projection valued measure from the characteristic functions for the half-openintervals [α,∞) at each L2(dµi, σ(A)), use the isomorphism with the irrep subspace Hi to obtain aprojection valued measure Pi([α,∞)) on Hi.

(b) Show, formally, that any function of A can be written as

f(A) =

∫σ(A)

f(α)dP (α) (76)

8 The spectral theorem for normal operators

We can now go through the program for the above derivation of the spectral theorem. We consider thecommutative C∗-algebra spanned by 1, A,A∗.

37

8.1 Gel’fand isomorphism

This has all been discussed at length: the algebra is nothing but the continuous functions on the spectrum.

8.2 A state defines a measure on σ(A)

A state — originally a positive linear functional on the algebra — can also be understood as a positivelinear functional on the continuous functions C(X) over the character set X of the commutative algebraby defining

ωX(fC) := ω(C), (77)

where C is an element from the Algebra and fC ∈ C(X) the corresponding function.Similarly, there is a continuous, positive linear functional on the continuous functions on σ(A): ωσ :

C(σ(A))→ C, ωσ(fC) = ω(C).The spectrum is a a compact set: it is a closed (the resolvent set is open) and bounded |a| ≤ ||A||

subset of the complex numbers. Therefore we can apply Riesz-Markov to see that there is a measure onσ(A) such that

ωσ(fσ) =

∫σ(A)

dµσfσ =

∫σ(A)

dµ(a)fσ(a). (78)

(The last being only a convention of notation.)

8.3 Construction of the L2(dµi, σ(A))

This is always the same trick: we do it for polynomials and then use the available norm to completethe space to make it Hilbert. Clearly, all polynomials on σ(A) appear as vectors of the GNS. We definean isomorphism between the vectors |P (A)〉 = P (π(A))|Ω〉 of the GNS representation and the functionsP (a) on σ(A). By the isomorphism, these are ∈ L2(dµi, σ(A)). The isomorphism extends to the norm-completion of either space by a standard procedure.

In this way we have obtained an isomorphism between the L2(dµi, σ(A) and our irreps on Hi and,putting then all together, the original representation on H =

⊕iHi.

8.4 Spectral theorem by projection valued measures

You will show in the exercises that one can formally write any hermitian operator as as an integral

πω(A) =

∫σ(A)

αdPω(α) (79)

with a projection valued measure on the spectrum Pω. We can restrict ourselves to irreps, where ω refersto a given pure state that characterizes our irreducible representation. The difference to the previous isthat we directly construct πω(A) ∈ B(H), rather than using the isomorphism: the map is already builtinto the construction. This technique will be used to construct the spectral representation for unboundedoperators, where we can no longer refer to the GNS constructions, that was all based on the C∗ properties,in particular the finite norm of the operator.

The integral is to be understood in the Lebesgue sense as∫σ(A)

αdPi(α) = sups

∑biPω(Mi), (80)

where the supremum is over the simple functions s(a) ≤ a.There are two points that one must take into consideration: first of all, we have not proven that we

actually can construct a projector valued measure from our state ω (except for the trivial projectors 0and 1). We have our measure by identifying the action of ω on elements of the algebra with an integral,but for elements in the algebra, which correspond the continuous functions c ∈ C(σ(A)). Projectors Pα

38

correspond to characteristic functions χ[α,∞), which are in general discontinuous and therefor are not inthe algebra. We can reach them by a limiting process, but not in the sense of the supremum-norm (afterall, our algebra is closed in this norm). Secondly, it is not obvious whether the integral is well-defined.

For both points we will use Lebesgue’s theorem about dominated convergence. The theorem estab-lishes convergence beyond the (supremum) norm-convergence using pointwise convergence instead:

Theorem 3. Dominated Convergence (Lebesgue) Let fn be a sequence of real-valued measurablefunctions on a measure space (S,Σ, µ). Suppose that the sequence converges pointwise to a function fand is dominated by some integrable function g in the sense that

|fn(x)| ≤ g(x)

for n all points x ∈ S. Then f is integrable and

limn→∞

∫S

|fn − f | dµ = 0

and also

limn→∞

∫S

fn dµ =

∫S

f dµ

Now, it is easy to construct a squence of continous functions cn ∈ C(σ(A) that converges to acharacterisitic function χM , M ⊂ Σ for any given x, “pointwise”. Also, the theorem guarantees thatthe integral exists. Then χM exists as a µω-measurable function in L2(dµω, σ(A)) and with that we candefine the projector-valued measure through

〈φ|P (Mi)ψ〉 :=

∫φ∗(a)χMi

(a)ψ(a)dµω(a).

The same reasoning can be applied to show that any positive function, then any real, finally anycomplex-valued function of a normal bounded operator can be defined as

f(A) :=

∫σ(A)

f(α)dP (α) (81)

We will come back to these ideas after the introduction of unbounded operators.

8.5 Quantum habits

In bra-ket notation of quantum mechanics, the above is

P ([α, β]) =

∫ β

α

dµac(γ)|γ〉〈γ|+∫ β

α

dµp(g)|g〉〈g| (82)

=

∫ β

α

dµac(γ)|γ〉〈γ|+∑

gi∈[α,β]∩σp(A)

|gi〉〈gj | (83)

(note that nobody usually considers any singular spectrum). The |γ〉〈γ| of σac are something like thederivative of a projection operator, formally

dP (γ) =dP

dγdγ (84)

As the derivative of a step-function, it has δ-character, and as you always suspected, the continuumfunctions are something like “

√δ(γ − γ′)”. Note however, that while we have met all instruments to

make things like dP (γ) household commodities, we have not remotely introduced the mathematics to

39

safely operate with the |γ〉. In fact, we really never need to do this, except for convenient formalmanipulations.

Problem 8.26: Limiting processes and measures Measures behave rather naturally underset operations and in limiting processes.

(a) Let (X,Σ, µ) be a measured space and Y ⊂ X a measurable set. Show that

ν(A) := µ(A ∩ Y ) (85)

defines a measure on X.

(b) Show that A ⊆ B ⇒ µ(A) ≤ µ(B) (monotonicity)

Let us now define limits of sequences of sets as follows

An A : An ⊆ An+1 and⋃n

An = A (86)

andAn A : An ⊇ An+1 and

⋂n

An = A (87)

(c) If An A then µ(An)→ µ(A) (continuity from above)Hint: Construct a series of distjoint sets such that

⋃nm=1Bm = An and compute the measures of

if the series.

(d) If An A then µ(An)→ µ(A) (continuity from below)Hint: Use the complement

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9 Bounded and unbounded operators in Hilbert space

9.1 Linear operator

Let H be a separable Hilbert space. A linear operator is a map

A : D(A)→ H (88)

where D(A) is a linear subspace of H called the domain of A. A is called bounded if

||A|| := sup||ψ||=1

||Aψ|| = sup||φ||=||ψ||=1

|〈φ|Aψ〉| <∞. (89)

9.2 B.L.T. theorem (bounded line transformation)

Let A : D(A) ⊂ X → Y be a bounded linear operator and let Y be a Banach space. If D(A) is dense,there a unique (continuous) extension of A to all of X, which has the same operator norm.

Problem 9.27: Proof the B.L.T. theorem for bounded operators in Hilbert space, whichsays, that any bounded operator A with a dense domain D(A) can be uniquely extended to an continuousoperator A on all of D(A) = H with the same norm.

(a) Construct such an extension explicitly by finding a “natural” map for those elements, that are notin the dense domain, and make sure you got a unique definition.

(b) Crosscheck linearity for the newly defined points of the operator.

(c) What is the reason the norms must agree?

(d) Explicitly convince yourself of the uniqueness.

9.2.1 Multiplication operator on L2(dx,R)

Multiplication by a measurable function can be defined as an operator with a dense domain D(A): Let

(Af)(x) := A(x)f(x), D(A) = f ∈ L2(dx,R)|Af ∈ L2(dx,R) (90)

and A(x) is a measurable function.For the proof the DA is dense, we use a procedure that is quite intuitive and will be used frequently

in the following: for any f ∈ H we construct a sequence of fn → f that is clearly in the domain. For thatwe multiply f by a charactaristic functions that exclude regions where the integral would become toolarge, the sets Ωn = x||A(x)| ≤ n. Ωn are Borel sets as we required A(X) to be measurable. ClearlyΩn → R such that Ωm ⊂ Ωn for m ≤ n: one writes Ωn R. The functions fn ∈ D(A) : Afn ∈ L2(dx,R).Also, ||fn − f || → 0 by Lebesgue dominated convergence, i.e. the fn are dense.

Next, one can easily see that A (considered as an operator) is bounded if the function A(x) is bounded.(More precisely, it should be essentially bounded: it may be as large as it wants, but only on sets of measure0. The correct way of capturing this idea there as a real number a the set Ia = x||A(x)| > a has measureµ(Ia) = 0.)

Problem 9.28: Bounded multiplication operator Let fA(x) be a measurable functionand A the multiplication operator on L2(dx,R)

(Af)(x) = fA(x)ψ(x), D(A) = ψ ∈ L2(dx,R)|Aψ ∈ L2(dx,R) (91)

Show that A is bounded iff fA is essentially bounded, i.e. there is a real number a such that Ia =x| |fA(x)| > a has measure 0.

41

(a) Show that ||A|| ≤ ||fA||∞, remember

||f ||∞ = infa|µ(Ia) = 0 (92)

(which adjusts the supremum-norm for the case of only essentially bounded functions).

(b) Show ||A|| 6< ||fA||∞ .Hint: By contradiction: if we had ||A|| = c, but c < ||fA||∞ we could construct a test functionφ ∈ H such that c < ||Aφ||that contradicts this.

It is good to remember why for an unbounded function the domain cannot be all of L2(dx,R):for example, in the case case A(x) = x functions behaving as |f(x)| ∼ 1/|x|1/2+ε, ε > 0 are certainly∈ L2, but the functions A(x)f(x) ∼ x/|x||1/2+ε are not. Another example is A(x) = |x|−1: any functionf ∈ L2(dx,R) behaving as |x|1/2−ε near x = 0 will certainly have a diverging integral. By a similarprinciple one can build functions f that are L2 for any A(x)f(x) that is not essentially bounded, i.e. thatexceeds any arbitrary bound on a set with non-zero measure.

Problem 9.29: Singular functions for unbounded multiplication operators Givena positive measurable function A : R → R+ that is not essentially bounded. Construct at least onefunction s ∈ L2(dx,R) for which ||As|| =∞.

9.3 Comments on the domain

A non-trivial dense domain is typical for unbounded operators. The existence of the domain also restrictsthe freedom for algebraic operations involving unbounded operators. For example, the domain of the sumof two operators D(A+B) is in general smaller then each of the domains of the pieces, we can choose

D(A+B) := D(A) ∩ D(B) (93)

9.4 Quadratic form and polarization identity

It is interesting to see that we do not need to know all matrix elements of an operator. It suffices to knowall its “expectation values” 〈ψ|Aψ〉. The map

qA : D(A)→ C : qA(ψ) = 〈ψ|Aψ〉

is called the quadratic form of the operator. We can construct all matrix elements of the operator fromits quadratic form by the polarization identity

〈φ|Aψ〉 =1

4[qA(φ+ ψ)− qA(φ− ψ) + iqA(φ− iψ)− iqA(φ+ iψ)]

9.4.1 Hermitian operator

A densely defined operator is called hermitian (= symmetric) if

〈φ|Aψ〉 = 〈Aφ|ψ〉 ∀φ, ψ ∈ D(A). (94)

Lemma 1. A densely defined operator A is symmetric if and only if its quadratic form is real-valued.

Problem 9.30: Hermitian operators and quadratic forms Show that a densely definedoperator A is hermitian if and only if its quadratic form qA is real-valued.Hint: Evaluate the quadratic form for a vector ψ + iφ

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9.5 Adjoint operator

Let A be an operator on a Hilbert space H defined on a dense subset D(A) ⊂ H. Then the adjointoperator A∗ is defined through

〈φ|A∗χ〉 = 〈Aφ|χ〉 ∀φ ∈ D(A). (95)

The domain D(A∗) are the vectors χ ∈ H for which sup||φ||=1 |〈Aφ|χ〉| < ∞. For bounded A, triviallyD(A∗) = H.

9.5.1 Comment on the domain D(A∗)

The operator A∗ is well-defined and its domain D(A∗) is determined once we have chosen D(A). Notethat in general D(A∗) 6= D(A∗). It may not even be dense, let alone all of H, or anything useful at all.

Example We can construct a δ-type operator on L2(dx, [−1, 1]) as

Aφ = φ(0),D(A) = φ continuous (96)

The χ ∈ D(A) means

supχ∈D(A)

∫ 1

−1

χ∗(x)φ(0)

||φ||<∞. (97)

As we can construct continuous L2 functions that are arbitrarily large at x = 0 (e.g. a δ-like sequenceof hat-functions), χ must be orthogonal to the constant function, i.e. the 1-dimensional subspace ofconstant functions gλ(x) ≡ λ 6∈ D(A∗).

9.6 Range and kernel

One can see that for densely defined A,

ran(A)⊥ = ker(A∗) : (98)

Let χ ∈ ker(A∗), i.e. A∗χ = 00 = 〈A∗χ|φ〉 = 〈χ|Aφ〉 (99)

which means χ ⊥ Aφ. On the other hand χ ⊥ Aφ implies the scalar product = 0 for a dense set of φ, i.e.A∗χ = 0.

9.7 Self-adjoint operators

The main technical distinction in the transition from bounded hermitian to self-adjoint operators is thatnot only the action of A and A∗ on some φ ∈ D(A) ∩ D(A∗) must be identical, but that it must bepossible to define D(A) such that also D(A) = D(A∗), only then we can write A∗ = A.

Any bounded hermitian operator can be made self-adjoint by extending its domain to D(A) = H,which is always possible by B.L.T.

For a hermitian operator

〈φ|Aφ〉 = 〈Aφ|φ〉 <∞ ∀φ ∈ D(A). (100)

Therefore, for a hermitian operator A, D(A∗) ⊇ D(A), we use the notation A∗ ⊇ A.Note that A∗ is not in general symmetric(=hermitian), even if A is symmetric.

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9.7.1 Multiplication operator

It is interesting to discuss this on the simple example of our multiplication operator. Let a multiplicationoperator A be as above and let us define

A : (Ah)(x) = A∗(x)h(x), D(A) = f∣∣ ||Af || <∞ (101)

Note that here we have (plausibly) chosen the domain D(A) and clearly D(A) = D(A). But is A = A∗?For the adjoint we cannot choose the domain, it is defined once A is fully defined. It is not a prioriobvious that there cannot be a h 6∈ D(A) such that |〈h|Afn〉| <∞∀fn ∈ D(A). To show that this cannothappen, let us assume h ∈ D(A∗). Then ∃g = A∗h such that

〈h|Afn〉 − 〈g|fn〉 = 0 ∀fn ∈ D(A). (102)

Now we use again the trick with restricting the integrations to a subset Ωn of R with finite measureΩn = x| |A(x)| < n, in which case fn = χΩnf ∈ D(A) for f ∈ L2, therefore also∫

h∗(x)A(x)fn(x)− (A∗h)(x)fn(x)dx = (103)∫[h(x)A∗(x)− (A∗h)(x)]∗χΩnf(x)dx = 0 ∀f ∈ L2, (104)

As Ωn ↑ R, this shows that (except possibly for sets of measure 0)

A∗(x)h(x) = (A∗h)(x).

on any h(x) ∈ D(A∗) the adjoint acts as multiplication by the complex conjugate function. By construc-tion of the adjoint, ||A∗h|| <∞ and therefore h ∈ D(A).

It follows immediately, that real-valued multiplication operators for measurable functions A(x) areself-adjoint, if we choose the domain as D(A) = f

∣∣||Af || <∞.9.8 Closed operator

If we have an densely defined unbounded operator we loose continuity. One typical way how unboundedoperators can behave is seen with the multiplication operators: as we approach with our functions fn ∈ L2

the places where A(x) diverges, the norms diverge ||Afn||. Somehow one feels that this is a rather benign,at least well understood failure of convergence. Somehow we are warned about what we are running intoas we approach the fringes of our domain. I we watch our steps and walk more slowly as ||Afn|| grows,nothing bad will happen. Contrast this with the situation where our domain just has a hole and nothingnear the hole indicates that there is a problem, like a δ-function.

The former case of unbounded operators is a very important one and justifies the following definitions

Definition 4. The graph ΓA of an operator A is the set of pairs ΓA = (ψ,Aψ)|ψ ∈ D(A).

Definition 5. The graph-norm of ψ ∈ D(A) for A is ||ψ||A := ||ψ||+ ||Aψ||.

Definition 6. An operator A is called closed, if D(A) is closed w.r.t. the graph norm || · ||A.

Note that, if the graph ΓA of A is closed as a subset of H ⊕H, the operator is closed: although thegraph norm ||φ||A is not identical to the norm

√||φ||2 + ||Aχ||2 defined on H ⊕ H, the two norms are

equvialent: if a sequence (φn, Aφn) ∈ ΓA ⊂ H⊕H Cauchy in either norm, it is also Cauchy in the other:

(||ψn||+ ||Aψn||)2 → 0⇔ ||ψn||2 + ||Aψm||2 → 0

any convergent sequence of Ψn = (ψn, Aψn) ∈ ΓA converges, if ||ψn||A = ||ψn||+ ||Aψn|| converges. If weknow that ΓA is closed w.r.t. the either norm, it is closed w.r.t. to the graph norm, i.e. the operator isclosed.

Closed operators are such that, if both a sequence ψn and the sequence Aψn converges, we know thatψn → ψ ∈ D(A). Typical cases where this can fail are δ-function like objects.

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Lemma 2. The adjoint operator A∗ of any A is closed.

This can be shown by a neat trick as follows.

Proof. In H⊕H we consider the unitary map J : (φ, χ)→ (−χ, φ). (for real vectors, this map is like anorthogonal rotation in that

〈(φ, χ)|J(φ, χ)〉 = 〈φ, χ| − χ, φ〉 = −〈φ|χ〉+ 〈χ|φ〉.)

The graph of the adjoint is the set of pairs of χ, η ∈ H ⊕H such that 〈χ, η|J(φ,Aφ)〉 = 0, i.e. 〈χ|Aφ〉 =〈η|φ〉 for all φ ∈ D(A). That means graph the (χ,A∗χ), χ ∈ D(A∗) consists of the pairs that areorthogonal to the subspaces J(φ,Aφ), φ ∈ D(A). We use that the orthogonal subspace of any subspaceis closed. Therefore the graph is closed w.r.t. to the H ⊕ H norm, which on the graph is equivalent tothe graph norm.

The discussion above suggests that operators that are defined everywhere and are closed must bebounded. That is indeed the case:Problem 9.31: Closed graph theorem Let A : H → H be a linear map that is defined on allH. A is bounded iff Γ(A) is closed.

Problem 9.32: Closed operator - counter-example Show that the operator (Aφ)(x) =φ(0) on D(A) = φ ∈ L2(dx, [−1, 1]), φ continuous is not closed.Hint: Construct a series such that (φn, Aφn) = (φn, 1), where φn → φ = 0 and compare the limit of thepairs with the pair at φ,Aφ.

9.9 Normal operator

We would also like think that of our multiplication operator as normal in the sense AA∗ = A∗A. Buthere we are in trouble, as even for real valued functions we do not a priory know what would be theoperator A∗A: what is the domain? This is not a trivial question, as, e.g., the domain for A(x) = xdefinitely differs from the domain for A2(x) = x2. We need to amend our previous definition “normal”:

Definition 7. An operator is called normal if D(A) = D(A∗) and ||Af || = ||A∗f || for f ∈ D(A).

Problem 9.33: Normal unbounded operators Bounded normal operators are defined byA∗A = AA∗. Because of the problems with defining the domain, for unbounded operators one callsA normal, if D(A) = D(A∗) and ||Af || = ||A∗f || ∀f ∈ D(A). Show that for bounded operators thiscoincides with our previous definition.

Problem 9.34: Normal operators are closed Show that.Hint: Use the lemma that A∗ is closed (will be proven in the lecture). The graph norms can be relatedusing the definition of normality.

9.9.1 The domain of self-adjoint operators is maximal

If we have defined an operator on a dense set and if we know it is self-adjoint, the domain is fixed andwe cannot extend it or reduce it without destroying self-adjointness.

Suppose there is B∗ = B ⊇ A = A∗. For any g ∈ D(B∗), by definition |〈g|Ah〉| < ∞∀h ∈ D(B) ⊇D(A), so we know g ∈ D(A∗), i.e. D(A∗) ⊇ D(B∗). As we assume that D(A) = D(A∗) as well asD(B∗) = D(B), we have

D(A∗) ⊇ D(B∗) = D(B) ⊇ D(A) = D(A∗)

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Caution this does not mean that all operators that are symmetric (but not self-adjoint), i.e. A∗ ⊃ Acan only be extended in one single way. This is definitely not the case: what you get to know as the“δ-potential” is obtained by a particular extension of the kinetic energy operator, functions ψ(0) 6= 0are removed from the domain, which destroys self-adjointness. From this point, different extensions arepossible, one of them being the “∆ + δ(x)”.

9.9.2 Closure of a symmetric operator

It is often easy to show that an operator is symmetric. It is often hard to show that it is self-adjoint.Maximality of selfadjoint operators can help to proof an operator is self adjoint. The domain of theadjoint operator is by definition maximal. For symmetric A it is larger than A∗ by definition of whatis symmetric: A∗ ⊇ A. The domain of the adjoint of the adjoint, D(A∗∗) is maximal. It contains allvectors f such that A∗∗f is a bounded linear functional. In particular, D(A∗∗) ⊇ D(A). When trying toextend an operator A to a maximal symmetric (self adjoint) operator, one can use that D(A∗) containsall vectors h for which 〈A∗h|f〉 = 〈h|A∗∗f〉 = 〈h|Af〉 exists, where f is from a dense set. We have therelations

A∗ ⊇ A, A∗ ⊆ A and also A∗∗ ⊇ A. (105)

that is, A∗∗ is “smaller” (or equal) than A∗, but it is “larger” (or equal) than A. Of course, A∗∗f = Affor f ∈ D(A). Therefore A∗∗ is an extension of A. It is called the closure A = A∗∗. It can happen thatA = A∗, in which case we call A essentially self-adjoint, implying that the self-adjoint extension of A(by closure) is uniquely determined.

9.9.3 The importance of self-adjointness for quantum mechanics

Popularity of self-adjoint operators is due to the fact that they (and only they!) have strictly realspectrum. This is convenient, if you want to directly associated a real number shown by some instrumentwith the spectral value.

However, the true importance is that there is a one-to-one relation between self-adjointness andunitary groups. Unitary groups strictly represent all differentiable symmetries and also, of course, thetime-evolution. No self-adjoint operator - no unitary evolution and loss of probability, conversely, ifthere as a unitary time-evolution, there is a Hamiltonian that is self adjoint. Similar things hold forany symmetry you might come up with: momentum, angular momentum, Lenz-Runge: the associatedconserved quantities MUST be represented by self-adjoint Hamiltonians. Naive manipulations with δ-likeobjects can lead to leaks of probability into Nirvana.

9.9.4 Example: the spatial derivative

Definition of domains (for essential self-adjointness) is related to boundary conditions for differentialoperators. Here we are at a point of great practical relevance: spectral properties, and with it the physicalmeaning of an operator vastly differ with the domain that we assign to it. The following examples mayappear contrived or obvious in hindsight, but actually, they are very strong illustration of the kind ofproblems associated with domain questions.

Let us compare the two operators on L2(dx, [0, 2π])

(A0f)(x)(Apf)(x)

= −i∂xf(x),

D(A0) = f ∈ C1([0, 2π])|f(0) = f(2π) = 0D(Ap) = f ∈ C1([0, 2π])|f(0) = f(2π) (106)

Both operators are symmetric, domains are dense with D(A0) ⊂ D(Ap). But only Ap is essentiallyself-adjoint, while A0 is not.

Appealing to physics knowledge, we can anticipate why: derivatives are (infinitesimal) translations.By exponentiating them, we should obtain macroscopic translations, but this cannot be for D(A0): wecannot push a non-zero function across the boundary without loosing the norm.

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Dirichlet boundary conditions, A0 We determine the domain of D(A∗). For any g(x) ∈ D(A∗0)there exists γ(x) ∈ L2 such that∫ 2π

0

dxg(x)(−i∂xf)(x) =

∫ 2π

0

dxγ(x)f(x), (107)

from which we conclude by partial integration that∫ 2π

0

dxf ′(x)[g(x)− i∫ x

0

dtγ(t)] = 0 ∀f ∈ D(A0) (108)

One can show (see problems) that this implies that g is “absolutely continuous”, i.e.

g(x) = g(0) +

∫ x

0

dtγ(t). (109)

We have established that A∗g = γ = ∂xg such that γ ∈ L2.These are two necessary conditions, but together also sufficient and thus they define the domain. The

two properties define the “first Sobolev space” H1, which is the domain D(A∗).

Problem 9.35: Absolutely continuous Show that∫ 2π

0

dxf ′(x)[g(x)− i∫ x

0

dtγ(t)] = 0 ∀f : f ∈ C1[0, 2π] and f(0) = f(2π) = 0

implies that g(x) is “absolutely continuous”:

g(x) = g(0) +

∫ x

0

dtγ(t). (110)

Note: In case you are wondering: there are functions c(x) that are continous, that are differentiablealmost everywhere (i.e. except on a set of measure zero), ∃∂xc, but are not absolutely continuous. Afamous example can be constructed using the Cantor set.

Problem 9.36: Absolutely continuous - counterexample We have seen that functionsf ∈ D(A∗0) of the adjoint of our differential operator

(A0f)(x) = f ′(x),D(A0) = f ∈ C1[0, 2π]|f(0) = f(2π) = 0

must be absolutely continuous, where C1 denotes the functions that are continuously differentiable almosteverywhere.

(a) Construct a continuous function that is not absolutely continuous c(x).Hint: Find a function with this property that is constant on the complement of the cantor set.Extend it to a continuous function that is defined everywhere on [0, 1]. (Cantor function)

(b) Show that this function is not from the domain of D(A∗0).

Is A0 essentially self-adjoint? Here we can illustrate the concept of closure and essential self-

adjointness: for essentially self-adjoint operators the uniquely defined A∗0?= A0 gives the self-adjoint

extension of the original A0. For the present case, we compute D(A). We choose any function f fromD(A0) ⊆ D(A∗),

0 = 〈g|A0f〉 − 〈A∗0g|f〉 = i(g∗(0)f(0)− g∗(2π)f(2π). (111)

where the second identity is established by partial integration. But for g ∈ D(A∗0) we have no restrictionon the boundary values and in particular we can have g(0) 6= g(2π). Therefore, f(0) = f(2π) = 0: also

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for the domain of the closure we must maintain the zero boundary conditions, i.e. A∗0 ⊃ A0, A0 is notessentially self-adjoint!

We do not have all info to extend A0 uniquely to a self-adjoint operator: we could, for example,use periodic boundary conditions, which leads to Ap, but we could just as well keep the domain “naileddown” at the boundaries, which would correspond to an extension by δ-functions. The resulting operatoris is no longer what we consider as a derivative ∂x.

Periodic boundary conditions, Ap Obviously Ap ⊃ A0, therefore A∗p ⊆ A∗0. We easily identify theadditional constraint on D(A∗p) by integration by parts shows

0 = 〈g|Apf〉 − 〈A∗pg|f〉 = [g∗(0)− g∗(2π)]f(0)⇒ g(0) = g(2π). (112)

which leads toD(A) = D(A∗∗) = f | ac , ||f ′|| <∞, f(0) = f(2π), (113)

i.e. first Sobolev space with periodic boundary conditions. We have the same necessary and sufficientconditions on D(A∗p) as on D(Ap), i.e. Ap is self-adjoint.

9.9.5 A criterion for self-adjointness

Lemma 3. Let A be hermitian. If there exists z ∈ C such that ran(A + z) = ran(A + z∗) = H, then Ais self-adjoint.

Proof. We will show that any ψ ∈ D(A∗) is also in D(a). Let ψ ∈ D(A∗) and denote A∗ψ = ψ. Sinceran(A+ z∗) = H, there is ϕ ∈ D(A) ⊆ D(A∗) such that (A+ z∗)ϕ = ψ + z∗ψ. Now

χ ∈ D(A), 〈ψ|(A+ z)χ〉 = 〈ψ + z∗ψ|χ〉 = 〈(A+ z∗)ϕ|χ〉 = 〈ϕ|(A+ z)χ〉

As this holds for all χ ∈ D(A) and ran(A + z) = H, we conclude ψ = ϕ and thus ∈ D(A), i.e. anyψ ∈ D(A∗)⇒ ψ ∈ D(A) ⊆ D(A∗).

Conversely, we see that if A is hermitian, but not self-adjoint, any point of one of the complex half-planes must be in the spectrum of A, i.e. A+ z or A+ z∗ or both are not invertible.

9.9.6 Spectrum of a self-adjoint operator

As for elements of a C∗-algebra, we define the spectrum as the complement of the resolvent set. We ask,more specifically, that the inverse exists as a bounded operator

ρ(A) = z ∈ C|∃(A− z)−1, ||(A− z)−1|| <∞ (114)

Using the above definition, we can see that the spectrum of a self-adjoint operator is strictly real:

1.||(A− z)φ|| = ||(A− x)φ+ iyφ|| ≥ |y| ||φ|| ∀φ ∈ D(A). (115)

i.e. ker(A− z) = 0, it is invertible for all y 6= 0.

2. For self-adjoint operators, we also have ker(A∗ − z∗) = 0. As ran(A − z)⊥ = ker(A∗ − z∗) weknow ran(A− z) is dense in H.

3. To see that it is not only dense, but all H, write ψ ∈ ran(A−z) as ψ = (A−z)φ, then ||ψ|| ≥ |y| ||φ||,then

||(A− z)−1ψ||||ψ||

≤ ||φ|||y| ||φ||

= |y| (116)

i.e. it is uniformly bounded for a dense subset, therefore also its supremum is bounded.

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4. Finally, we know that for any ψ ∈ H there is a squence φn : Aφn → ψ. As self-adjoint operatorsare closed, the limit φ,Aφ is in the graph, i.e. φ ∈ D(A).

Note: Self-adjointness is needed here, as for χ ∈ D(A∗)\D(A) we cannot use the above argument forbounding ||(A∗− z∗)χ||, as we need to use 〈A∗χ|χ〉 = 〈χ|A∗χ〉 = 〈χ|Aχ〉 ∀χ ∈ D(A∗), which we are notallowed to do, if we only know our operator is symmetric. Note that the non-zeroness is not on a densesubset is not sufficient to conclude ker = 0.

Counter-example The operator D = −i∂x on [0,∞) with zero boundary condition is hermitian.However, its adjoint has the eigenvector e−x with eigenvalue −i. Therefore ker(D∗+ i) 6= 0 and ran(D−i) 6= H (not even dense). Therefore D + i is not invertible and i ∈ σ(D).

• Is there an eigenfunction to the eigenvalue i? Does not need to be, for example also for x there isno sensible “eigenfunction”.

• Is there a sequence of functions fn ∈ D(D) such that ||(D − i)fn|| → 0? No, not in this case,as actually the complete lower half plane is in the spectrum and we cannot approach the singularpoint from non-singular points. We can only approach the boundaries of the resolvent set in thisway. For self-adjoint operators, any spectral point is a boundary point of the resolvent set.

• Again, in hind sight from Stone’s theorem, this is not surprising: the derivative would shift functionsto the left, but we cannot shift them to negative x, they are just not in our Hilbert space.

Problem 9.37: Hermitian, not self-adjointNote: There were a few misprints in this problem.Let A be the operator on H = L2(dx, [0,∞))

(Af)(x) = (i∂xf)(x), D(A) = φ ∈ H|φ ∈ C1, φ(0) = 0 (117)

(a) Show that A is hermitian

(b) Determine D(A∗)

(c) Show that all z, =m (z) < 0 are eigenvalues of A∗ and determine the eigenfunctions.

(d) Conclude that σ(A) includes the whole upper half plane.Note: We had only loosely introduced the resolvent set as the z where (A − z)−1 “exists as abounded operator”. It was implied that it is at least densly defined.

9.10 Quadratic forms

There is a feeling that, if only we know all possible expectation values of an operator, i.e. if we knowthe associated quadratic form q, we know the operator. We had seen this in the “duality” between“observables” and “states”. So, given a complete set of “expectation values” A(ψ), i.e. given a quadraticform with a dense domain for ψ, can we find an operator A, such that E(ψ) = 〈ψ|Aψ〉?

Yes, under some circumstances (Teschel Theorem 2.14). A general quadratic form derives from asesquilinear form on a Hilbert space s(φ, ψ) as q(ψ) := s(ψ,ψ). (It is the “diagonal” of the sesquilinearform.) The connection to a self-adjoint operator can be established, if the quadratic form has a lowerbound, i.e. ∃γ ∈ R such that E(ψ) ≥ γ,∀ψ ∈ Q(E) and if it is closed: Q is closed with the form norm||ψ||q = ||(A− γ)ψ||+ ||ψ||.

Theorem 4. To every closed semi-bounded quadratic form q defined on Q there corresponds a uniqueself-adjoint operator A such that for its quadratic form qA = q with Q = QA, where

ψ ∈ QA : qA(ψ) = 〈ψ|Aψ〉 <∞

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Proof. (find it in Teschl, Theorem 2.14)

Note: Typical Schrodinger operators are semi-bounded: there is a lowest energy. However, note thatin elementary quantum mechanics you meet the Hamiltonian for the Stark effect, with a potential ~r · ~E ,wich has no lower bound, with a lot of mathematical trouble as a consequence.

Note: A typical quadratic form that is not closed is the δ-potential

δ(ψ) =

∫dxψ(x)∗δ(x)ψ(x) = |ψ(0)|2 : (118)

tent-functions with their peak at x = 0 approach the 0-function, also the value of the quadratic form isconstant, i.e. it is a Cauchy-sequence under the form norm, but the limit under the form norm 6= 0.

10 The spectral theorem for self-adjoint operators

We have formulated the spectral theorem so far only for bounded normal operators. We will now extendthis to unbounded self-adjoint operators. We go through the following steps:

• We recall the spectral theorem for bounded operators in the formulation using projection valuedmeasures.

• Given any projection valued measure, show we can construct, for any decent (=measurable) functionf an operator P (f). That operator will be normal. If f is unbounded, P (f) is unbounded. Themain point is to get the domain right.

• We will circumvent the problems with unboundedness by considering the family of bounded operatorsRA(z) = (A− z)−1 (Resolvent!).

• We will appeal to measure theory to state (without proof) that there exists a unique measure µψ(λ)such that

〈ψ|RA(z)ψ〉 =

∫R

(λ− z)−1dµψ(λ) (119)

• From this measure, one deduces a projection valued measure that allows formulating the spectraltheorem exactly as for bounded operators.

10.1 Bounded spectral theorem with projection valued measure on RFrom now on we only treat self-adjoint operators, not the wider class of normal operators. We haveestablished that a commutative algebra of bounded hermitian operators that are part of a C∗ algebra areunitarily equivalent to the continuous functions on the spectrum. The spectral representation of anysuch operator can be expressed as

A =

∫σ(A)

λdP (λ) (120)

for some projection valued measure P (λ) of the half-intervals (−∞, λ]. One can define the expectationvalue

〈ψ|P (λ)ψ〉 =

∫Rχ(−∞,λ](α)dµψ(λ). (121)

In the original representation we had not Hilbert space given a priory, and the main difficulty was thatχ(−∞,λ] is not a continuous function, therefore it is not obvious, whether there exists such a P (λ).

With the Hilbert space given, one can prove the existence of P (λ) as a projector as follows.

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Lemma 4. Let H be a Hilbert space and µψ be the measure on R corresponding to the state ωψ(A) =〈ψ|Aψ〉 on the C∗-algebra of the bounded operators on H. Then

〈ψ|P (λ)ψ〉 =

∫Rχ(−∞,λ](α)dµψ(λ). (122)

defines a projection operator on H.

Proof. We construct a sequence of continuous functions cn(t) that converge pointwise to the characteristicfunction cn(t) ↑ χ(−∞,λ](t). This lets us define the multiplication operators Cn and a sequence of quadraticforms

qn(ψ) = 〈ψ|Cnψ〉 =

∫Rcn(t)dµψ(t). (123)

For fixed ψ, qn(ψ) is Cauchy and we denote the limit by q(ψ). q is bounded from below as all q(ψ) =lim qn(ψ) ≥ 0. It is closed: it is defined everywhere and the form norm ||ψ||q = ||ψ|| + q(ψ) < 2||ψ||is equivalent to the norm on H. Therefore there exists an operator P (λ). By construction, P (λ) is thestrong limit of the Cn. It is bounded, as Cn ≤ 1. What remains to show is the projector property. Aswe now know that all operators are bounded, showing the property in the strong limit is sufficient

〈ψ|(P 2 − P )ψ〉 = 〈ψ|(C2n − Cn)ψ〉+ ε→ 0 (124)

which can be shown by cutting from the integration the ranges Ωnε = t|cn(t)2 − cn(t) > ε, for whichclearly µψ(Ωnε )→ 0.

10.2 Resolution of identity P (λ)

We abstract the above properties of the P (λ) into the following definition:

Definition 8. A resolution of identity P (λ) := P ((−∞, λ]) with the properties

1. P (λ) is an (orthogonal) projection operator P 2(λ) = P (λ) = P ∗(λ).

2. P (λ1) ≤ P (λ2) for λ1 ≤ λ2

3. P (λn)s→ P (λ) for λn ↓ λ (strong right-continuity)

4. P (λ)s→ 0 for λ→ −∞ and P (λ)

s→ 1 for λ→∞ and

Lemma 5. To any projection valued measure there corresponds a resolution of identity. Conversely, anyresolution of identity uniquely defines a projection valued measure.

• Obviously, the projection valued measures of the intervals (−∞, λ] have these properties. Also,this can be turned around: given a resolution of identity, there is a projection valued measure andrelation is unique (we do not proof this here).

• We need to restrict to right continuity as our measure may contain point-measures: thinking of acharacteristic function χ(−∞,λ] with σ(λ) 6= 0: if we approach this from above, it is counted intothe measure all the time. If, in contrast, we approach λ from below, we would have a sudden jumpas we include the last point λ and cannot expect left-continuity.

10.3 Construction of operators from projection valued measures

We have discussed that the integral

P (f) :=

∫Rf(λ)dP (λ) (125)

defines an operator for bounded continuous functions and for the characteristic function. Obviously, thisextends to bounded measurable functions (i.e. any function for which Lebesgue-integration works).

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P (f) for unbounded measurable f We choose the domain

Df = ψ ∈ H|∫R|f(λ)|2dµψ(λ) <∞, (126)

where µψ(λ) = 〈ψ|P (λ)ψ〉 is a positive measure on R. It is easy to see that this is a linear subspace.

Problem 10.38: Show that Df is a (open) linear subspace and that it is dense.Now, by ever the same trick, we construct a sequence of bounded functions converging to f by

multiplying by the characteristic functions

fn = χΩnf, Ωn = λ| ||f(λ)| < n (127)

which is Cauchy in L2(dµψ,R) as f ∈ L2. Then also P (λ)fn = ψn is Cauchy w.r.t. H: for any projectorP : ||Pψ|| < ||ψ||. Therefore we can define

P (f)ψ := limn→∞

P (fn)ψ, ψ ∈ Df . (128)

Theorem 5. For every Borel-measurable function f the operator

P (f) =

∫Rf(λ)dP (λ), D(P (f)) = Df (129)

is normal and satisfies

||P (f)ψ||2 =

∫R|f(λ)|2dµψ(λ), 〈ψ|P (f)ψ〉 =

∫Rf(λ)dµψ(λ) (130)

for every ψ ∈ Df .

For bounded f complex conjugation, multiplication, and addition of functions can be mapped ontothe operators as follows in more or less obvious ways:

1. P (f)∗ = P (f∗)

2. Linearity, with domains adjusted to D(αf + βg) = D|f |+|g|

3. Product, with domain adjusted as D(fg) = Dg ∩ Dfg

Proof. We first show that P (f)∗ = P (f∗): We start from bounded functions fn = fχΩn , clearlyD(P (f)) = Df = Df∗ and

〈φ|P (fn)ψ〉 = 〈P (f∗n)φ|ψ〉

for φ, ψ ∈ Df = Df∗ , which extends to the limit n → ∞ by continuity. In particular φ ∈ Df∗ ⇒ φ ∈D(P (f)∗) or D(P (f)∗) ⊇ Df .

Remains to show D(P (f)∗) ⊆ Df . The property P (fn) = P (f)P (Ωn) follows from the constructionof P (f). With ψ ∈ D(P (f)∗) and any φ ∈ H we have

〈P (f∗n)ψ|φ〉 = 〈ψ|P (fn)φ〉 = 〈ψ|P (f) P (Ωn)φ︸ ︷︷ ︸∈D(P (f))

〉 = 〈P (Ωn)P (f)∗ψ|φ〉. (131)

As we know that P (f∗)ψ ∈ H for ψ ∈ D(P (f)∗, we can write

||P (f∗)ψ|| = limn→∞

||P (Ωn)P (f∗)ψ|| = limn→∞

∫χΩn |f∗|2dµψ =

∫|f |2dµψ, (132)

i.e. ψ ∈ Df .Normality trivially follows by inserting ||P (f)∗ψ|| = ||P (f∗)ψ|| = ||P (f)ψ||.

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10.3.1 The subspace HψWe now repeat the construction used for Hilbert space representations of a C∗ algebra using the resolutionof identity and the map P (f), but taken the Hilbert space as given and defining the state as ωψ(A) =〈ψ|Aψ〉.

Definition 9. We denote by Hψ ⊆ H the closed subspace

Hψ = P (g)ψ|g ∈ L2(R, dµψ) (133)

Hψ is a closed subspace, as the L2 is closed and a sequence P (gn) converges if and only if gn converges.As P : g → P (g)ψ conserves the norm, ||g||L2 = ||P (g)ψ||Hψ , the existence of a limit in L2 implies thatP (g)ψ ∈ Hψ and vice versa.

We recognize the construction similar to the one used during GNS and expect that Hψ is in somesense an invariant (not necessarily irreducible) subspace under the action of P (f). However, as we admitunbounded functions now, we need to deal with domain questions again and we cannot as easily reasonby purely algebraic arguments. Instead, we introduce the following:

Lemma 6. The subspace Hψ =: PψH. reduces the operator P (f); i.e. PψP (f) ⊆ P (f)Pψ.

Remember: A ⊆ B means D(A) ⊆ D(B), Af = Bf ∀f ∈ D(A). This expresses that the operators P (f)“commute” with the projector Pψ onto the subspace Hψ at least for functions from Df . It will allow usto define the operator restriced to the subspace PψiP (f)Pψi , as in the unbounded case.

Proof. For f bounded it is easy, as we can use commutativity of multiplication without worrying aboutdomains. We study the action of the operators on any φ by decomposing it into its part in Hψ and theorthogonal complement. Let φ ∈ Df = H, then there exists g ∈ L2(R, dµψ) such that φ = P (g)ψ + φ⊥.Clearly, for any h ∈ L2(dµψ) we have 〈P (h)ψ|P (f)φ⊥〉 = 〈P (f∗h)ψ|φ⊥〉 = 0 as f∗h ∈ L2. As this holdsfor all h, we know P (f)φ⊥ ∈ H⊥ψ .

PψP (f)φ = PψP (f)(P (g)ψ + φ⊥) = Pψ P (fg)ψ︸ ︷︷ ︸∈Hψ

= P (fg)ψ = P (f)P (g)ψ (134)

For unbounded, we first use the standard trick of restricting to functions fn from the domain con-verging to f . For every such bounded function we can reason as above and we find for φ ∈ Df

P (fn)Pψφ = P (f)P (Ωn)Pψφ→ PψP (f)φ. (135)

However, it is not clear whether P (f)Pψφ ∈ Df , even if φ ∈ Df . After all, we are cutting from thefunction φ only the piece that is in Hψ. Here we use that P (f) is normal and therefore closed: asP (f)φn = P (f)P (Ωn)φ is convergent, the limit φn → φ ∈ D(P (f)).

By that we can decompose any operator P (f) into PψP (f)Pψ ⊕ P (f)⊥: because of the projectorproperties and the “commutation” there are no “cross-terms”: φ ∈ Hψ and χ ∈ H⊥ψ

0 = 〈χ|PψP (f)φ〉 = 〈χ|P (f)Pψφ〉 = 〈χ|P (f)φ〉 (136)

10.3.2 Unitarity, cyclic vectors, spectral basis

The map between functions g ∈ L2(R, dµψ) and function P (g)ψ ∈ Hψ is unitary, i.e. linear, bijectiveby construction and it conserves the norm. We call ψ cyclic if Hψ = H. For cyclic vectors, we havegained a representation of an arbitrary (possibly unbounded) operator P (f) on Hψ = H, that is unitarilyequivalent to a representation of P (f) as multiplication on L2(R, dµψ). For non-cyclic vectors, we iteratethe procedure as in the C∗ algebra case until the (countably many) vectors of H are exhausted. Asequence of such vectors ψi is called spectral basis

All in all, given an algebra of now unbounded functions on R, we have re-established properties as wehad seen it in the bounded case. What is left is to find the projection valued measure that will correspondto the possibly unbounded self-adjoint operator A and a given ψ.

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10.4 For every projection valued measure there is a self-adjoint operator

We choose for a(λ) = λ the operator P (a) =: A (with domain Da) is self-adjoint: it is manifestlyhermitian and we have shown that P (a) is normal, i.e. D(A) = D(A∗). Also, for this particular operator,the construction above gives its “spectral representation” in the proper sense, i.e. on L2(dµψ) it acts asmultiplication by its spectral values.

10.5 For every self-adjoint operator there is a spectral measure

We are almost done. We have seen that, if there is a resolution of identity P (λ), we can obtain operatorsin H and, for the operator P (a) we already have its spectral representation. The last step is to finda spectral measure dµψ and the corresponding resolution of identity such that the operator acts as themultiplication by λ for the Hilbert space over R with that measure: L2(R, dµψ). We will appeal here toa few facts from measure theory.

10.5.1 The resolvent of P (a)

Let a(λ) = λ be the identical function and denote A = P (a). For complex z, the operator

RA(z) =

∫R

(λ− z)−1dP (λ) (137)

is the resolvent of A. The corresponding quadratic form

Fψ(z) =

∫R

(λ− z)−1dµψ(λ) (138)

is called the Borel transform of the measure µψ. This function clearly maps the upper half plane ontoitself and is analytic on the upper half plane.

10.5.2 A measure is uniquely defined by its Borel transform

Interestingly, there is a one-to-one relation of functions as above and measures on R:

Theorem 6. Let the function Fψ(z) be any function that is holomorphic in z and maps the upper half-plane onto itself (Herglotz). Then it is the Borel transform of a uniquely defined measure µψ:

Fψ(z) =

∫R

1

λ− zdµψ(λ) (139)

The measure µψ is finite ∫µψ <∞.

It can be explicitly constructed by the Stieltjes inversion formula.

µψ(λ) = limδ↓0

limε↓0

1

π

∫ λ+δ

−∞=(Fψ(t+ iε))dt (140)

Remark Me make no attempt on proving this. We only point out that that limit is frequently encoun-tered in the form

limε↓0

1

π=(

1

x− x0 + iε

)= δ(x− x0). (141)

The limit δ ↓ 0 just makes sure that we approach the final projectors in a continuous way, as we onlyhave right-continuity.

Problem 10.39: Stieltjes inversion formula

54

(a) Show that

limε↓0

1

π=(

1

x− x0 + iε

)= δ(x− x0). (142)

Hint: Recall the meaning of∫dxf(x)δ(x − x0) and compute the above limit. Use the residual

theorem for an integration path along a circle around x0.

(b) Show, formally, the Stieltjes inversion formula

µψ(λ) = limδ↓0

limε↓0

1

π

∫ λ+δ

−∞=(Fψ(t+ iε))dt, (143)

where Fψ is the Borel transform of µψ:

Fψ(z) =

∫R

(λ− z)−1dµψ(λ)

10.5.3 The resolvent defines a measure dµψ

By the theorem above, we obtain a spectral measure µψ, that we had constructed more directly in thecase of bounded operators.

Lemma 7. For a self-adjoint operator A the function

Fψ,A(z) = 〈ψ|(A− z)−1ψ〉. (144)

is Herglotz.

Problem 10.40: Herglotz property Let A be self adjoint and

Fψ,A(z) := 〈ψ|(A− z)−1ψ〉.

(a) Show that Fψ(z) is analytic in a vicinity of any point z0 with =m (z0) 6= 0.Hint: Expand (A− z)−1 around z0 using the resolvent formula.

(b) Show that for =m (z) > 0 also =m (Fψ(z)) > 0

Note: We had shown that for self-adjoint operators σ(A) ⊆ R. This is equivalent to saying that theresolvent of A

RA(ψ) = (A− z)−1ψ (145)

exists as a bounded operator for =(z) 6= 0 and is defined on all of H. Now consider the function

Fψ,A(z) = 〈ψ|(A− z)−1ψ〉. (146)

This function is analytic in the upper half plane: one can show this using the resolvent formula for z ina sufficiently small surrounding of z0. Because self-adjointness of A, it maps the upper half-plane ontoitself: let z = x+ iy and φ ∈ D(A) : (A− z)φ = ψ

=(〈ψ|(A− z)−1ψ〉) = =(〈(A− z)φ|φ〉) = 〈−iyφ|φ〉 = iy〈φ|φ〉. (147)

Therefor there exists a measure µψ such that this function is its Borel-transform. Therefore, by theStieltjes inversion we obtain a spectral measure µψ.

Note: Analyticity of matrix elements of the resolvent is a very useful property: it allows us to, forexample, change contours in the integration and, in the end, defines the resolvents themselves as beinganalyic in z.

55

10.5.4 A resolution of identity P (λ)

We have stated earlier (without proof) that a semi-bounded and closed quadratic form corresponds to aself-adjoint operator. Indeed, the functional

q(−∞,λ](ψ) =:

∫Rχ(−∞,λ](t)dµψ(t) (148)

is non-negative (i.e. semi-bounded). Closedness

qλ(ψ) =

∫ λ

−∞dµψ ≤

∫ ∞−∞

dµψ(λ) <∞ (149)

Therefore an operator P (λ) exists, whose quadratic form is qλ.Next one shows that the P (λ) form a resolution of identity; here, as we are dealing with bounded

functions throughout, no substantially new questions arise and we skip the details.

10.5.5 The spectral projectors

Lemma 8. The family of operators PA(Ω) defined by

〈φ|PA(Ω)χ〉 =

∫RχΩ(λ)dµφ,χ(λ) (150)

forms a projection valued measure.

10.5.6 Spectral theorem for self-adjoint operators

To every self-adjoint operator A there corresponds a unique projection valued measure PA such that

A =

∫RλdPA(λ) (151)

This theorem is central to all we do in QM. Among others, it allows us to define functions of a s.a.operator as

f(A) =

∫Rf(λ)dPA(λ). (152)

This definition of functions f(A) has the virtue of clearly settling the domain questions and being ap-plicable a to a wide class of functions, including discontinuous functions like the characteristic function.Even constructions like “δ(A− a)” can be given meaning and we will used them in scattering theory.

11 Stone’s theorem

It was mentioned that there is a one-to-one relation between self-adjoint operators and unitary groups likethe time-evolution or continuous symmetry transformations. The latter have the much more immediatephysical interpretation and justify why we insist to working with self-adjoint “observables” in quantummechanics.

11.1 A self-adjoint operator generates a unitary group

Theorem 7. Let A be self-adjoint and let U(t) = exp(−itA) Then

1. U(t) is a strongly continuous one-parameter unitary group.

56

2. The limit limt→01t [U(t)ψ − ψ] exists if and only if ψ ∈ D(A) and it holds:

limt→0

1

t[U(t)ψ − ψ] = −iAψ (153)

3. The domain is invariant under U(t) : U(t)D(A) = D(A).

A one-parameter unitary group is a family of unitary operators U(t1 + t2) = U((t1)U(t2), U(0) = 1

(from which U(t)−1 = U(−t) = U∗(t)). Strongly continuous means that for tn → t, U(tn)s→ U(t).

Proof. • 1, group property: the standard exponential has this property, so in the spectral represen-tation we can read off the group property

• 1, strong continuity:

limtn→t||e−itnAψ − e−itAψ||2 = lim

tn→t

∫R|e−itnλ − e−itλ|2dµψ(λ)

=

∫R

limtn→t

|e−itnλ − e−itλ|2dµψ(λ) = 0.

We can exchange the limit with the integral as we have dominated convergence, i.e. point-wiseconvergence and domination by an integrable function, viz. the constant function 4 ≥ |e−itλ−e−itλ|2.

• 2, for ψ ∈ D(A) the limit exists

limtn→0

||e−itnAψ − e−itAψ + iAψ||2 = limtn→0

|1t(e−itλ − 1) + iλ|2dµψ(λ) = 0 (154)

• 2, if the limit exists, ψ in the domain: we will show that any operator where the limit exists, issymmetric. Then, as a self-adjoint operator is maximal, any such operator coincides with A. Denoteby A an extension of A on the domain of all functions where a limit A exists, then A is symmetric:

〈φ|Aψ〉 = limtn→t〈φ| i

t[U(tn)− 1]ψ〉 = lim

tn→t〈−it

[U(−tn)− 1]φ|ψ〉 = 〈Aφ|ψ〉 (155)

But self-adjoint operators are “maximal”, i.e. A = A.

• 3: invariance: a simple consequence of the commutative group property: if the limit exists for ψ,i.e. ψ ∈ D(A), then it also exists at U(s)ψ

||[

1

tU(t)− 1

]ψ|| = ||U(s)

[1

tU(t)− 1

]ψ|| = ||

[1

tU(t)− 1

]U(s)ψ|| (156)

One can also show that this is the only solution to the Schrodinger equation.

11.2 Stone theorem - unitary groups derive from self-adjoint operators

This inversion of the above is physically more interesting: we expect that the time-evolution should becontinuous in time and that it conserves the norm. It is not completely obvious why it would be linear.The latter one can argue if we want the time-evolution that cannot lead to a decrease of von Neumannentropy. This is plausible, as the simple time-evolution should not change the probability of a givenstatistical mixture. If we accept the premises above, the Schrodinger equation is a consequence.

Theorem 8. (Stone) Let U(t) be a weakly continuous one-parameter unitary group. Then it has theform U(t) = exp(−itA) for some self-adjoint A.

57

We can here use “weaker” conditions on U(t): as the U(t) are bounded, weak continuity implies strongcontinuity. (Proof can be found in the literature, we have not discussed some of the rather basic tools todo that proof.)

Proof. • A is densely defined. For now, we do not know any domain, and picking any ψH could makethe limit fail. We construct a certain “fuzzyness” w.r.t. to the time-evolution into our state, suchthat we can be certain that the limit exists.

• Take any ψ ∈ H and define

ψτ :=

∫ τ

0

U(t)ψdt (157)

(the integral is well-defined as U(t) are bounded), which implies 1τ ψτ converges to ψ, i.e. the ψτ

are dense.

• We can do the limit for the ψτ : the special construction allows us to interchange the roles of U(t)with U(τ), such that the limit τ → 0 can be taken:

1

t[U(t)ψτ − ψτ ] (158)

=1

t

∫ t+τ

t

U(s)ψds− 1

t

∫ τ

0

U(s)ψds (159)

=1

t

[∫ t+τ

0

. . .−∫ t

0

. . .−∫ τ

0

. . .

](160)

=1

t

∫ t+τ

τ

U(s)ψds− 1

t

∫ t

0

U(s)ψds (161)

= U(τ)1

t

∫ t

0

U(s)ψds− 1

t

∫ t

0

U(s)ψdst→0−→ U(τ)ψ − ψ. (162)

The dense set of ψτ is contained in D(A). Clearly, the limit must be symmetric and the domain isinvariant under the action of the group, as before.

• ker(A∗ − z∗) = 0 for =(z) 6= 0: suppose this were different, i.e. there is an eigenfunctionA∗φ = z∗φ. Then for ψ ∈ D(A) we can construct an exponentially growing matrix element:

d

dt〈φ,U(t)ψ〉 = 〈φ,−iAU(t)ψ〉 = −i〈A∗φ,U(t)ψ〉 = −iz〈φ|U(t)ψ〉 (163)

〈φ|U(t)ψ〉 = exp(−izt)〈φ|ψ〉. But the lhs is bounded, therefore the only choice is 〈φ|ψ〉 = 0 for allψ ∈ D(A).

• Need to skip: the fact that the adjoint of a symmetric operator has complex eigenvalues is thetypical indication that either we cannot extend the symmetric operator to a self-adjoint operatoror that the extension is not unique. Conversely, if there are no such eigenvectors, the symmetricA is “essentially self-adjoint”, i.e. its closure A = A∗∗ is self adjoint and this extension is unique.(Without proof: if you are looking for the proof, check for “defect indices” or “Caley transform”).

• We can now compare U(t) with V (t) = exp(−itA). We denote a possible difference as ψ(t) =[U(t)− V (t)]ψ for ψ ∈ D(A). One calculates

lims→0

ψ(t+ s)− ψ(t)

s= iAψ(t) : (164)

We find the norm of ψ(t) does not change in time

d

dt||ψ(t)||2 = 2<〈ψ(t), iAψ(t)〉 = 0, (165)

58

and as at time t = 0 there is no difference, there is never a difference, i.e the time-evolutions agreeon D(A), and as U(t) is bounded, the extension to V (t) is unique.

Problem 11.41: Weak and strong continuity Weak convergence of a sequence ψn ψmeans

〈χ|ψn〉 → 〈χ|ψ〉 ∀χ ∈ H. (166)

(a) Let ψn ψ. Show that ψn → ψ if lim ||ψn|| = ||ψ||.

(b) Let B(t) a family of unitary operators that is weakly continuous at t = 0, i.e.

limt→0〈φ|B(t)χ〉 = 〈φ|B(0)χ〉 ∀φ, χ ∈ H. (167)

Show that this implies strong continuity at t = 0: limt→0 ||B(t)χ|| = ||B(0)χ||

Problem 11.42: Stone theorem Complete the proof of Stone’s theorem. We had constructed adomain where the limit limt→0(U(t)− 1)/t = A is essentially self-adjoint. Denote by V (t) = exp(−itA),where A is the closure of A. Show that U(t) = V (t).Hint: Show that the evolutions U(t) and V (t) agree for ψ ∈cD(A) and remember B.L.T.

12 Further important topics

12.1 Self-adjoint Hamiltonians

We know that −∆ is self-adjoint, what happens if we add a potential? One finds that if a s.a. operator Ais modified by a symmetric operator B that is not “too big” compared to A the result is again selfadjoint.This the content of the following

Theorem 9. (Kato-Rellich) Let A be selfadjoint and B with D(B) symmetric. If there are constants0 ≤ α < 1 and 0 ≤ β such that

||bψ|| ≤ α||Aψ||+ β||ψ|| ∀ψ ∈ D(A), (168)

then A + B, D(A + B) = D(A) is self-adjoint. If A is essentially selfadjoint, then A + B is essentiallyself-adjoint with A+B = A+B.

The proof is simple IF we use the same fact that we used in the proof of Stone: (A ± iη)D(A) = Hfor some η ⇔ A selfadjoint, i.e. neither half-complex plane is in the spectrum of A∗.

Proof. First we find a bound for the norm of the resolvent ||(A ± iη)−1|| ≤ |η|−1 we had done thatbefore (e.g. problems09), it is easy to see in the spectra representation of the selfadjoint A and clearly||A(A± iη)|| < 1. Then we write

(A+B ± iη)D(A) = [1 +B(A± iη)−1]︸ ︷︷ ︸I

(A± iη)D(A)︸ ︷︷ ︸II

. (169)

The space II is = H (or dense), as A is selfadjoint (or essentially s.a.), the first part has ker(I) = 0 asB(A± iη) < 1 by the relative boundedness of B: with ψ = (A− iη)φ, ||ψ|| = ||(A− iη)φ||

||B(A± iη)−1ψ|| =≤ α||Aφ||+ β||φ|| < α||A(A± iη)−1ψ||+ βη−1||ψ|| < ψ (170)

for sufficiently large η. Therefor ker(1−B(A± iη)−1) = 0.

59

12.2 Self-adjointness of typical Hamiltonians in Physics

We know that the Laplacian can be defined as a self-adjoint operator. We know for sure in the Fourierspace, where the Fourier transform F(−∆) ∝ ~k2 is a multiplication operator. The ~k2 is a continuousfunctions therefore measurable and with the domain

D(~k2) = φ(k)|∫d(3)k|~k|4|φk|2 <∞ (171)

it is self-adjoint.The Fourier-transform is defined on all L2(R3, dr(3)) and invertible (even unitary, i.e. norm-conserving),

therefore the Laplacian is s.a. on the domain

D(−∆) = χ|χ = F(φ), φ ∈ D(~k2). (172)

That rather obvious fact can also be proven by considering that −∆ is quite obviously symmetric andusing our criterion of self adjointness (A± i)D(A) = H:

F−1(∆± i)D(∆) = F−1(∆± i)FF−1D(−∆) = (~k2 ± i)D(~k2) = H,

where the last equality follows from ~k2 being s.a.

12.2.1 Domain of essential self-adjointness

The domain D(∆) is not very accessible in practice. Here the concept of essential self-ajointness playsout its use: we can rather easily define a domain De(∆) essential self adjointness

De(∆) = C∞c (R3), (173)

i.e. the infinitely differentiable functions with compact support. The closure of this is the self-adjointLaplacian as defined above:

(∆,De) = (∆,D). (174)

We leave this without proof.

12.2.2 Self-adjoint −∆ + V

For a large class of potentials the Hamiltonians on L2(d(3)r,R3) are self adjoint. These are all potentialsof the form

V = w + v, w ∈ L2(R3), v ∈ L∞. (175)

This can be proven directly using Kato-Rellich with the help of estimates that are typical tools offunctional analysis. We will use a different approach that has further reaching implications. In theproof, we have seen that it is important to estabilish invertibility of 1 + B(A ± iη)−1 to obtain allH = (−∆ + V ± iη)D(∆). One has the feeling that, if only B(A± i) is bounded, one can always increaseη until ||B(A± iη)−1|| < 1. This is captured in the

Definition 10. A hermitian operator B ⊇ A is called relatively bounded to A if the map B : DΓ(B)→ His continous w.r.t. the the graph norm of || · ||A.

A more transparently, one formulates this as

Corrolary 1. B is bounded relative to A iff there exists M such that

||Bφ|| ≤M(||H0φ||+ ||ψ||) ∀φ ∈ D(A). (176)

60

We will prove this property for the Coulomb potential. Connecting to Kato-Rellich will be left as anexercise.

Problem 12.43: Relatively bounded Show that the following three properties are equivalentfor s.a. operators A and B : D(B) ⊇ D(A):

1. B : DΓ(B)→ H is continous w.r.t. the the graph norm of || · ||A.

2. There exists M such that

||Bφ|| ≤M(||Aφ||+ ||ψ||) ∀φ ∈ D(A). (177)

3. The operator B(A− z)−1 is bounded for at least one z.

12.3 Perturbation theory

Perturbation theory is based on the idea that if we add to an “unperturbed” self-adjoint H0 a perturbationαH1 that we can make sufficiently small by α→ 0, the spectrum and eigenvectors will depend analyticallyon α such that an expansion for small α will approach the exact result. The problem is that, in infinitedimensional Hilbert spaces, the idea of what is “small” needs to be made precise. For example, the dipoleinteraction ~E ·~r of an atom with an electric field is small for some states (the ones localized close to zero),but is arbitrarily large for states at a large distance. Conversely, the Coulomb potential is very largeat the origin, but rather weak at large distances. The key is, as above for Kato-Rellich, to look at thestrength of H1 relative to H0.

12.4 Analyticity of eigenvalues

We study the analyticity of the resolvent Rz(α) := (H0 + αH1 − z)−1 as function of α. We can rightaway write the formal power series

(H0 − z + αH1)−1 = (H0 − z)−1∑k

αk[H1(H0 − z)−1

]k. (178)

The rhs will norm-converge for all sufficiently small α, if H1(H0−z)−1 is bounded. The above proofs that

Lemma 9. Let H1 be relatively bounded w.r.t. H0, i.e. H1(H0− z)−1 is bounded for =m (z) 6= 0. Thenthe resolvent (H0 +αH1−z)−1 is analytic w.r.t. to α and z in an open set of C×C including 0×ρ(H0).

12.4.1 Riesz projections

It is very useful to consider the following representation of projectors onto spectral subspaces

Lemma 10. Let E0 an isolated eigenvalue of a selfadjoint H and γ(E0) ⊂ ρ(H) a counterclockwiseclosed path in the resolvent set that encloses E0 and but no other spectral values. The operator

− 1

2πi

∫γ(E)

dz(H − z)−1 = PE0(179)

projects onto the subspace of eigenvectors to the eigenvalue E0.

Note: by “isolated” we mean that there is such a γ(E0). The proof using the residual theorem istrivial if we invoke the spectral theorem.

Problem 12.44: Riesz projection Define an eigenvalue with finite multiplicity as a spectralvalue E0 ⊂ σ(H) of a s.a. H such the associated projector valued measure P (E0) is a finite-dimensional

61

projector. We call E0 isolated if there exists and open set U such that U ∩ σ(H) = E0. Show theproperties of the Riesz-projections.

This is rather useful as it shows the intimate relation between spectral projectors (and with it thespectral theorem) and the resolvent. Resolvents are in general easier to work with than the originaloperator as they are bounded operators and the domain conundrums can be avoided. Obviously, byencircling more than just one eigenvalue, we obtain projections onto larger parts of the spectrum.

Analytic functions are known to “survive” integrations: let f(z, x) be functions that are analytic in asurrounding U of z0 for all x ⊂ γ. Here γ(t) describes some integration path in the complex plane. Forsimplicity, assume that c is compact and f(z, x) is bounded uniformly for all x ∈ γ. Then the integral

F (z) =

∫dtf(z, γ(t)) (180)

is analytic w.r.t. z.So if we can show analyticity of (H0 + αH1 − z)−1, we have analyticity of projectors and with this

of the eigenvalues. By analyticity of a bounded operator we mean that it has a norm convergent seriesexpansion in terms of bounded operators, i.e. it can be written as

P (α) =∑k

αkP (k) (181)

where all P (k) are bounded.

Theorem 10. Let H1 be bounded relative to H0. Then the isolated eigenvalues of H = H0 + αH1 withfinite multiplicity and the projectors onto them are analytic w.r.t. α in an open surrounding of α = 0.

The main (not very big) complication for the proof is that both, the spectrum and the spectralprojectors “move” at the same time, i.e. both depend on α. This, by the way, is not only a mathematical,but also the main practical complication, that makes higher order perturbative formulae so ugly. Strategyis to first show that the spectral projectors are analytic. Next compute a unitary transformation to anoperator, where the vectors do NOT depend on α and show that the eigenvalues depend analytically onα.

Proof. • As the resolvents exist and are analytic as functions of z for any sufficiently small range ofα, we can consider

P (α) = − 1

2πi

∫γ

dz(H0 + αH1 − z)−1, (182)

where γ now includes z such that the resolvents remain analytic for all |α| < α0. Such an environ-ment exists by the above lemma.

• We can now replace the integrand by its power series in α and z and perform the integrations w.r.t.z, thus analyticity of P (α) is proven.

• Note that P (α) has the same finite rank for α in the range of analyticity: We know that it is aprojector. For projectors dim(P (α)H) = Tr(P (α)). By asumption, Tr(P (0)) is finite. We knowthat Tr is continuous w.r.t. to the norm and as an analytic function the norm ||P (α)|| is continuousw.r.t. to α. As the rank is a discrete quantity it cannot change its value in a continuous way.

• For given α we constrain the operator to the subspace of the eigenvectors

Hc(α) = P (α)H(α)P (α). (183)

This is obviously an operator of finite rank. We will show that it is unitarily equivalent to anoperator that acts on the fixed subspace P (0)H.

62

• For that we writePc(α) = W (α)Pc(0)W ∗(α). (184)

The W (α) are operators that map from the unperturbed eigenspace to the perturbed eigenspace.Formally, such a W (α) : Pc(0)H → P (α)H is

W (α) = Pc(α) [1 + P (0)(Pc(α)− P (0))P (0)]−1/2

Pc(0) (185)

The inverse exists as Pc(α) − P (0) is arbitrarily small, for the same reason the square root existsand W (α) is analytic. The property (184) can now be verified by algebraic manipulations.

• The map W ∗(α) : Pc(α)H → Pc(0)H conserves the norm: suppose ψα from P (α)H, then

||ψα||2 = 〈ψa|P (α)|ψα〉 = 〈ψaW (α)P (0)W ∗(α)ψα︸ ︷︷ ︸∈P (0)H

|| = ||W ∗(α)ψα||2 ∀ψα.

and as the dimensions of the spaces agree, W (α) is invertible. As a map between the finite-dimesional spaces W (α) is unitary: isometry+invertible=unitary.

• NowHc(α) = W (α)P (0)W (α)∗H(α)W (α)P (0)︸ ︷︷ ︸

H(α), on P (0)H

W ∗(α) : (186)

what we have gained is that we can investigage the finite-dimensional matrix H(α): all functionsinvolved are analytic, and for the finite-dimensional H(α) this implies that also the eigenvalues areanalytic functions of α. Because of unitarity of W (α) they are identical to the eigenvalues of Hc(α).

Problem 12.45: Perturbation theory

(a) Show that for the projectors onto the unperturbed and perturbed sub-spaces it holds

Pα = WαP0W∗α.

for sufficiently small α with

Wα = Pα [1 + P0(Pα − P0)P0]−1/2

P0

Instructions: Multiply from the left by W ∗α and show W ∗αWα = 1, similarly for the other side. Forthe algebra, use [. . .]P0 = P0[. . .] and evaluate PαP0(P0PαP0)−1P0Pα. by writing the projectors as∑k |k, x〉〈k, x|, x = 0, α. Also verify and use P0[. . .]P0 = P

(b) Show that map on finite-dimensional spaces W (α) : P (α)H → P (0)H is unitary.

63

13 Scattering theory

Literature: Reed and Simon, Vol III: Scattering Theory; Teschl, Mathematical Methods in QuantumMechanics.

13.1 Scattering is our window into the microscopic world

For example Rutherford experiment, COLTRIMS (Cold Target Recoil Ion Momentum Spectroscopy),and many more. One rare non-scattering microscopic method: Scanning tunneling microscope.

Detectors can register: direction of emission, kinetic energy of arriving particle, possibly its internalstate (spin, excitation state, mass), arrival times. Arrival times are often used to determine kinetic energyfrom knowing time of emission and distance to the point of arrival (“time of flight” detector).

13.1.1 Cross section

Given an ensemble of incident sub-system/particles characterized by ~ki, si, i = 1, . . . , I with momenta~ki, internal states si (e.g. spin, constituents, excitation state etc., if any), and flux Fi (number of suchsubsystems per time per surface), one measures the number of sub-systems/particles emitted per timeNe into momenta and internal states ~qe, se, e = 1, . . . , E. The cross section is the ratio

σ(~k1, s1, . . .~kI , sI ; ~q1, s1, . . . ~qE , sE

)=NeFi

(187)

This is what we ultimately need to know for comparison with observation.

13.1.2 Potential scattering

All deeper mathematical discussion in these lectures will be limited to the case of a single particle thatscatters without change of its internal state from a fixed, immutable target that is represented by apotential. I.e. 1 = I = E and si = se, such that the only relevant two arguments of the cross section arethe initial and final momenta ~ki and ~qe. Physicists call this “potential scattering”.

Further we will not treat a statistical ensemble of particles but rather a particle described by a singlewave function. This is a minor technical simplification useful at the present level.

13.2 Mathematical modeling of a (simple) scattering experiment

Prepare a particle in some “free” state, e.g. free wave packet

Φ(~r,−t) = (2π)−3/2

∫d(3)kei

~k·~rψ(~k) (188)

at some time −t. Prepare the packet such that no part of it is in the range of interaction.Let that wave packet evolve according to the scattering time-evolution. Wait long enough, say, until

t, and analyze the wave packet outside the range of the interaction.The fundamental mathematical issues are:

• What does “preparing the state” mean? Can we actually map any arbitrary free wave packet intoa scattering wave packet? Is that mapping unique? (Existence and uniqueness of scatteringsolutions.)

• What is the fate of the scattering state? Can the system just “swallow” an incoming particle? Orcan we be certain to find the system again in a (possibly quite different) free wave packet at somelarge time t? (“Asymptotic completeness”)

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13.3 The Møller operators

Now assume that far back in time a given wave packet can escape the action of the potential. At thattime, free and interacting time evolution are indistinguishable for the wave packet. Now we can considerthe wave packet as an interacting one and evolve forward in time to the present by the full time evolution.Under which circumstances can we find times large enough to do such a mapping? Do the following limitsexist:

limt→±∞

U(t)U0(−t) := Ω±? (189)

In which sense may the limits exist? Can it exist in an operator sense? Given ε,

?∃T : ||U(t)U0(−t)− Ω±|| < ε, ∀ ± t > T (190)

(“Uniformly” for all states): probably not, we only need to choose arbitrarily slow wave packets, i.e.particles that move extremely slowly out of the interaction region at a speed v < L/T , where L is somecharacteristic dimension of the interaction region. However, it may exists “pointwise” for any wave packet

||[U(t)U0(−t)− Ω±]|Φ〉|| → 0 for ± t→∞ (191)

That is, the time-limit can exist in the sense of a strong limit.

13.3.1 Properties

Note that the Møller operators map a wave-packet composed of free electrons (plane waves, exp(i~k~r) into

a wave packet composed of scattering solutions Ψ(±)~k

(~r) with asymptotic momenta ~k. We will find that

the energies remain unchanged in the process (as it should be). We expect, in some sense,

Ψ(±)~k

(~r) = Ω±ei~k~r. (192)

and

−1

2∆ei

~k~r = H0ei~k~r =

~k2

2ei~k~r and (H0 + V )Ψ

(±)~k

= HΨ~k =~k2

2Ψ~k (193)

We say “in some sense”, as the involved waves are not in Hilbert space.For the absolutely continuous spectrum we expect an intertwining property of the Ω±, i.e.

HΩ± = Ω±H0. (194)

13.4 The Lippmann-Schwinger (LS) equation

Frequently, the first form how a physics student is exposed to scattering theory is in the form of the

Lippmann-Schwinger equation for the Ψ(±)~k

:

|Ψ(±)~k〉 = lim

ε↓0|ei~k~r〉 − 1

H0 − (k2/2± ıε)V |Ψ(±)

~k〉. (195)

This is an equation for “scattering eigenfunctions” (± for “in/out”)

H|Ψ(±)~k〉 =

k2

2|Ψ(±)~k〉 (196)

with the HamiltonianH = H0 + V (197)

H0 the “free” Hamiltonian, V = H −H0 the interaction (potential), and Φ. . . “scattering eigenfunction”of H0

H0Φ = EΦ (198)

65

Usual choice

H0 = −1

2∆ (or :−∆) (199)

The quotation marks indicate that the “eigenfunctions” are not in the Hilbert space and therefor theirexistence and meaning need to be related to the operators that are defined as linear operators in Hilbertspace.

In position space the Lippmann-Schwinger equation is an integral equation

Ψ(±)(~r) = eı~k·~r − 1

4π

∫d(3)r′

e∓ı|k||~r−~r′|

|~r − ~r′|V (~r′)Ψ(±)(~r′) (200)

It is not at all obvious how this equation is related to the above description of a scattering exper-iment. How, for example, would we extract a cross section from Ψ(±)? And why would it be relatedto the time-dependent definition of scattering given above? And how do Ψ(±) fit into our Hilbert spaceformulation of quantum mechanics? Later in this introduction we will by formal manipulations derivethe equation. Historically, of course, it derives from the well studied scattering of classical waves. Here,for the conceptually clear definition of scattering theory we use the limit of standard time-dependentquantum mechanics to large times, as one does effectively in an experiment, where detection occurs atmacroscopically large times and at macroscopically large distances after a microscopic scale reaction.

Problem 13.46: Inverse of the Møller operator In the derivation of the Lippmann-Schwinger equation from the definition of the Møller operators we have used abstract reasoning to inter-change φ↔ ψ(±). Instead, directly show

Ω−1± = [1 + (H0 − k2 ± iε)−1V ] (201)

13.4.1 In- and Out-states

Physically, we easily can imagine “outgoing” waves, i.e. waves that from a given finite time onwards will(in some sense) move away from the scattering center. Similarly “ingoing” waves up to some finite time,always move towards the scattering center. The signs in the Lipmann-Schwinger equation refer to thesetwo physical situations, but it is not completely obvious how to cast that idea into a clear mathematicalconcept. In the end, we will be able to do this by looking at ~x · ~k: states ψ(t) where these are positivewill be “outgoing”, if negative ingoing.

It is important to note that in- and out-going are concepts of the long time limit, or, putting itdifferently, they are related to the behavior of the solution at large distances. At close range, the conceptis not meainingful. What seems to be outgoing at one point may revert direction under the influence ofsome force and become ingoing. Also, there is no possibility to characterize in- our outgoing by spectralproperties. For stationary states, as they appear in the LS equation, the total flux is conserved, so clearlythe in- and out-going parts some will be equal. An unabigous way to think about that concept will beintroduced in the following.

13.5 Spectral properties and dynamical behavior

In physics we like to think of bound states - that never seriously leave some finite region, and unboundstates, that are certain to leave any finite surrounding of the system. We also have got accustomed toassociating these two types of states with discrete and continuous spectrum, respectively. Not much ofa justification is given for this ususually. We will first establish that fact rigorously. The main theoremexpressing this fact is called RAGE (for Ruelle, Amrein, Georgescu, Enß).

13.5.1 Spectral subspaces

Given the separation of the spectral measure into point, absolutly continous and singular spectrum

〈φ|dP (λ)ψ〉 = dµφ,ψ = dµp;φ,ψ + dµac;φ,ψ + dµsing;φ,ψ (202)

66

and the fact that the three parts are mutually singular (Lebesges decomposition theorem) we find thatwe can decompose the Hilbert space into mutually orthogonal spaces

H = Hp ⊕Hac ⊕Hsing =: Hp ⊕Hc. (203)

We have lumped singular and ac into “continuous” Hc, as these two part behave similar in many respectsand actually, usual Schrodinger operators have empty singular spectrum. We omit the comparativelysimple proof to this fact. We may also write

Hp = span(eigenvectors) (204)

andHc = H⊥p (205)

Functions ψ ∈ Hac behave like wave packets, i.e.

limt→±∞

〈e−itHψ|φ〉 = 0, (206)

we can say: after long time, the wave packet becomes orthogonal to any fixed state.For functions from the sing spectrum a similar decay only holds in the time-averaged sense. This kind

of spectrum is poorly suitable for scattering considerations, as we cannot exclude that the system returnsto its initial state, even if only for a short time. Note that Hamiltonians with sing spectrum are notnecessarily pathological cases: they may possibly be realized in quasi-crystals (i.e. with quasi-periodicpotentials). We will not consider them for our discussion of scattering theory.

For proving long-time decay of continuous packets, we first state

Theorem 11. (Wiener) Let µ be a finite complex Borel measure on R and let

µ(t) =

∫Re−iλtdµ(λ)

be its Fourier transform. Then the Cesaro time average of µ(t) has the limit

limT→∞

1

T

∫ T

0

|µ(t)|2dt =∑λ∈R|µ(λ)|2,

where the sum on the right hand side is finite.

Note that the sum is over individual points, not any finite intervals: the theorem somehow tells usthat on average the Fourier transform of the square reduces to a the measure of a discrete set of points.One thing we readily observe is that here the ac and sing parts of the measure cannot contribute as theyare zero on discrete points. Basically, as we will see in the proof, the averaging over t (“time”) in thelimit T →∞ generates δ-like functions.

Proof. As our measure is finite and the functions are measurable, we can apply Fubini’s theorem toevaluate

1

T

∫ T

0

|µ(t)|2dt =1

T

∫ T

0

∫R

∫Re−i(x−y)tdµ(x)dµ∗(y)dt

=

∫R

∫R

1

T

∫ T

0

e−i(x−y)tdtdµ(x)dµ∗(y)

The t-average converges point-wise to χ0(x − y) and it is bounded, such that (by “dominated conver-gence”) we have

=

∫R

∫Rχ0(x− y)dµ(x)dµ∗(y) =

∫Rµ(y)dµ(y) =

∑y∈R|µ(y)|2

67

Remember similar manipulations that you will have seen in your quantum mechnics course: actually,only the long-time limit establishes sharp, well-defined energies and only bound states can realize thesein a physical system.

Problem 13.47: Spectral subspaces We define the projectors onto spectral subspaces of Aas P (xx) = χxx(A) with xx = p, sing, or ac, Hxx = P (xx)H and χxx the characteristic functions of themutually disjoint subsets Mxx.

(a) Reassure yourself of the decomposition of H = Hp ⊕Hac ⊕Hsing

(b) Show that Hxx are invariant under the time evolution e−itA.

(c) Show that Hxx reduces A.

13.5.2 Fourier transform of smooth functions (Riemann-Lebesgue)

We had discussed the absolutely continuous measures as those that are equivalent to the Lebesgue measurein the sense dµac(x) = f(x)dx with an integrable function f . That means that for the absolutelycontinuous part of the spectral measure we can take advantage of the following

Theorem 12. (Riemann-Lebesgue) If Lebesgue integral of |f | is finite, then the Fourier transform of fsatisfies

f(λ) :=

∫Rf(x) exp(−iλx) dx→ 0 as |λ| → ∞.

We leave this important and useful theorem without proof. It makes precise our intuition that a highlyoscillatory integral (|z| → ∞) will eventually average to zero, if the integrand is otherwise “smooth”, i.e.without non-integrable singularities.

13.5.3 Compact operators

We have discussed bounded operators, and often they behave just as operators on a finite dimensionalspace. But they are still acting in an infinite dimensional space and, when acting as potentials, thecan induce infinitely many (if not infinitely large) changes. For example, adding the (relatively) boundedpotential to the Laplacian, (like, for example the Coulomb potential) we can create infinitely many boundstates. Or, even more trivially, adding λ1 to an operator shifts the complete spectrum by λ.

First, we define genuinely “finite” operators:

Definition 11. An operator of finite rank is an operator whose range is finite dimensional. Thedimension of the range is called the rank of the operator

rank(K) = dim ran(K).

The general form of a compac operator is

Kψ =

n∑j=1

φj〈χj |ψ〉, (207)

where the φj are an orthonormal basis for ran(K). Operators that can be norm-approximated by finite-rank operators are called compact operators: We call K compact if there exists a sequence of finiterank operators Kn : Kn → K (in the sense of the operator norm).

In a situation as above, such operator can add only a finite number of bound states to the Laplacianand it cannot shift the continous spectrum. (We ommit the proof at this point.)check

Finally, what will be important for us is not that the operator itself is compact (e.g. the Coulombpotential is not), but that it is compact relative to some other operator (e.g. the Laplacian). Theconstruction is a for relative bounded:

68

Definition 12. An operator K is compact relative to a s.a. operator A if D(K) ⊇ D(A) and K(A−z)−1

is compact for some z 6∈ R.

Theorem 13. Let H0 be s.a. and H1 D(H1) ⊇ D(H0) relatively compact, i.e. the operator H1(H0−z)−1

is compact. Thenσess(H0 +H1) = σess(H0). (208)

The essential spectrum σess is all but the isolated point spectrum. I.e. it is σac ∪ σsing plus possiblelimit points of the point spectrum. For illustration, think of the Hydrogen atom, where 0 is a limit pointof the point spectrum.

A very useful property of compact operators is that they turn a weakly convergent sequence into astrongly convergent one.

Lemma 11. Let K be compact, then

ψn 0⇒ ||Kψn|| → 0. (209)

To understand why this happens, remember that the typically weakly convergent but not norm-convergent sequence ψn was the one that escaped to ever new directions. A compact operator is near-zeroon all but a finite subspace, i.e. on all but a finite set of directions. So the ψn at one point will be turningto directions where the compact operator just beats them down.

Problem 13.48: Compact operators Let K be a compact operator and φn φ, ||φn|| < c.Prove that ||K(φ− φn)|| → 0

[HVZ Theorem (Hunziker, van Winter, Zhislin, atomic Hamiltonian): existence of a lower bound,stability of matter.]

13.5.4 The RAGE theorem

Relative compactness allows us to single out a “finite” subspace of in the spectral range of an operator.The following theorem tells us, that no wave packet from the continuous spectrum can ever be containedin such a finite subspace in the long run.

Theorem 14. Let A be self-adjoint and K be relatively compact and denote by P (c)H = Hc and P (ac)H =Hac the projectors onto the continuous and absolutely continuous spectra, respectively, then

limT→∞

1

T

∫ T

0

||Ke−iAtP (c)ψ||2dt = 0, limT→∞

||Ke−iAtP (ac)ψ||2 = 0

for every ψ ∈ D(A). If in addition K is bounded, then the result holds for every ψ ∈ H.

The theorem tells us: if we evolve a continuous wave packet it will leave the finite space where Kdiffers significantly from A. Note that the norm ||e−iAtP (c)ψ||2 is constant, but its evolves into ever newdirections. If ψ is build from the absolutely continuous spectrum, that holds strictly. If ψ may containsingular spectrum, there may be moments in time where the norm “revives” to be sizeable, but theseperiods will be short.

Proof. Denote ψ ∈ Hc or ∈ Hac.

• For any finite rank operator the property is a direct consequence of Wiener’s theorem: Remember

||Kψ(t)||2 = ||n∑j=0

φj〈χj |ψ(t)〉||2 =

n∑j=0

|〈χj |ψ(t)〉|2

(the φj were chosen to be orthonormal)

69

• Denoting (in the spectral rep)

µ(t) =

∫dλe−iλt〈φj |P (λ)ψ〉 =

∫dλe−iλtdµφj ,ψ(λ)

we recognize the situation of Wiener’s theorem. As we have restricted ourselves to Hc, the point-measure is zero. For the ac spectrum, we can directly refer to Riemann-Lebesgue.

• As sum over the χj is finite, the statements are valid for the sum as well.

• For K compact, choose a sequence of finite rank operators Kn → K such that ||Kn −K|| < 1/n.Then

||Kψ(t)||2 ≤(||Knψ||+

1

n||ψ||

)2

≤ 2||Knψ||2 +2

n||ψ||2

So the theorem holds for K compact.

• For the general case, choose any ψ ∈ D(A). As P (c)A = P (c)A (more precisely: Hc reduces A,ψ ∈ Hc if and only if ψ = (A− i)−1φ and φ ∈ Hc. With

Ke−iAtP (c)ψ = Ke−iAtP (c)(A− i)−1φ = K(A− i)−1︸ ︷︷ ︸compact

e−iAtP (c)φ

we are back to the previous situation.

• If K is bounded, we approximate D(A) 3 ψn → ψ ∈ H and obtain the estimate

||Ke−iAtψ|| ≤ ||Ke−iAtψn||+ εn||K||.

This theorem allows us to formulate the following theorem, that tells us that time-evolution separatesthe (at least on average) the Hilbert space into parts that leave any finite subspace and parts that willremain contained in a finite subspace.

Theorem 15. (RAGE). Let A be self-adjoint. Suppose there is a sequence of relatively compact operatorsKn which converges strongly to identity. Then

Hc = ψ ∈ H| limn→∞

limT→∞

∫ T

0

||Kne−itAψ||dt = 0

Hp = ψ ∈ H| limn→∞

supt≥0||(1−Kn)e−itAψ||dt = 0

This theorem tells us that we can characterize Hc and Hp completely by the behavior of functionsunder the long-time limit of quantum mechanical evolution: the first part says that the fact that wavepackets composed of the continuous spectrum will be “diluted” out of any “finite” subspace. “Finite”means the space where K differs relevantly from A (relatively compact). In physics terms: if a wavepacket fully spreads (in the time-average), then it completely belongs to the continuous spectrum.

The second part tells us that if we can find a “finite” space (in the sense as above) where the wave-packet remains contained, the wave packet belongs purely to the continuous spectrum: spectral propertiesand time-evolution behavior match one-to-one.

Proof. Denote ψ(t) = exp(−itA)ψ, start with first part:

• Let ψ ∈ Hc a function from the contiuous subspace, then For using the very similar the previoustheorem we need to get a square into the integral. We do this by Cauchy-Schwartz (with a secondfunction g(t) ≡ 1):

1

T

∫ R

0

||Kψ(t)||dt ≤

(1

T

∫ T

0

||Kψ(t)||2dt

)2

→ 0.

70

• If ψ 6∈ Hc, i.e. ψ = ψc + ψp. If we we find that then for some large n that ||Knψp(t)|| ≥ ε > 0, wehave made our point.

• In fact, we can even claimlimn→∞

supt≥0||Knψp(t)− ψp(t)|| = 0.

• We write ψp(t)∑j aj(t)ψj for orthonormal eigenfunctions. (Remember the ψp(t) remains in Hp.

Truncate this after N terms. Now we have a fixed subspace and by assumption of the theorem theKn converge strongly, i.e. the statement holds for the truncated function.

• We need to refer to an auxiliary lemma (Teschel Lemma 1.14, essentially the Banach-Steinhaus/uniformboundedness) to infer from strong convergence of the Kn uniform boundedness: ||Kn|| < M . Thisallows to extend the statement from the finite sum to the complete ψ.

• For the second equation, first consider ψ ∈ Hp. From what we have just shown, (1−Kn)ψ(t)→ 0

• Finally, assume Hp 3 ψ = ψp + ψc. We are done, if we find that ||(1 −Kn)ψc(t)|| does not tendto 0 with n. We show, that we obtain a contradiction if it were to tend to 0 with n. The also theintegral tends to 0, i.e. for given ε < ||ψc|| there is an n such that

ε > limT→∞

1

T

∫ T

0

||(1−Kn)ψc(t)||dt

≥ limT→∞

1

T

∫ T

0

∣∣ ||ψc(t)|| − ||Knψc(t)||∣∣dt

≥ ||ψc(t)|| − limT→∞

1

T

∫ T

0

||Knψc(t)||dt = ||ψc|| > 0.

Problem 13.49: Corrolary of RAGELet A be self-adjoint, Kn a sequence of relatively compact operators converging strongly to the identity

(these are the same Kn as for RAGE theorem). For ψ in the Hilbert space H = Hp ⊕ Hc (Recall thedecomposition of the Hilbert space into the pure point and the continuous subspace) define

P pψ = limn→∞

limT→∞

1

T

∫ T

0

eitAKne−itAψdt, (210)

P cψ = limn→∞

limT→∞

1

T

∫ T

0

eitA(Id−Kn)e−itAψdt. (211)

Prove, that P p projects onto Hp and P c onto Hc.

13.6 The Møller operators

Definition 13. Let H and H0 be two self-adjoint opertors. The wave operators (Møller operators) aredefined as

D(Ω±) = ψ ∈ H|∃ limt→±∞

eitHe−itH0ψ

Ω±ψ = limt→±∞

eitHe−itH0ψ

We call D(Ω±) the set of outgoing/ingoing states and ψ± the states with ougoing/ingoing asymptotics.

71

Lemma 12. The sets D(Ω±) and ran(Ω±) are closed and Ω± : D(Ω±)→ ran(Ω±) is unitary.

Note that the roles of H and H0 can be interchanged, it is our choice which of the two time-evolutionswe consider primary. It is just the fact that with H0 = −∆ we know the time-evolution very well thatfavors the conventional choice.

Proof. • Clearly, ||Ω±ψ|| = || limt→±∞ eitHe−itH0ψ|| = limt→±∞ ||eitHe−itH0ψ|| = ||ψ||

• D(Ω± closed: let ψn ∈ D(Ω±), Cauchy, then Ω±ψn is Cauchy and the limit exists, i.e. ψn → ψ ∈D(Ω±).

• ran(Ω±) closed: we interchange the roles of H ↔ H0 to see that ran(Ω↔± ) = D(Ω±), therefore alsoran(Ω±) is closed.

• By the above we also see invertibility of the Ω±.

Note that we use a slightly more general definition of unitary as norm-conserving bijection.

13.7 S-matrix

As a first step for connecting to experimental observables, we ask: what is the probability of finding asystem, that in the remote past was a free wave packet state χi, say with a narrow momentum wave-packet around a given ~k, in the free momentum wave packet χe around ~q after scattering. For thatwe map the two wave free wave packets into the actual scattering solutions φi and φe by applying thescattering operators Ω+ and Ω−, respectively. Note that we choose two different signs, as we want tomap the ingoing free wavepacket χi to the scattering wave packet φi = Ω−χi, such that they are equalat the remote past, and conversely φe = Ω+χe with mapping in the remote future.

By general quantum theory, our probability is

〈ϕi|ϕe〉〈ϕe|ϕi〉. (212)

All we need to know is

〈ϕe|ϕi〉 = 〈Ω+χe|Ω−χi〉 = 〈χe|Ω∗+Ω−χi〉 =: 〈χe|Sχi〉 (213)

The operator S := Ω∗+Ω− = Ω−1+ Ω− is traditionally called S-matrix and obviously contains all information

that we are interested in for a scattering experiment. Note that at this point of the discussion theadjoint Ω∗+ is not well defined. Only after showing asymptotic completeness of the Ω± we can identify

P (H)(ac)H ≡ P (H0)(ac)H and thus define the standard adjoint. The scattering matrix S is then a unitarymap between the ac spectral parts of the free time evolution S : Pac(H0)→ Pac(H0).

Note that in this reasoning the relation to the cross section is still a bit murky, as we do not clearlyspecify at which point of their evolution we compare the two scattering wave packets. In the limit ofinfinitely narrow wave packets, this question disappears: in this limit, the particles become completelydelocalized and it does not matter, at which time we compare the wave packets, the particles are “alwayseverywhere”. We will not explicitly go through the rather tedious exercise of actually taking the limit.

13.7.1 Asymptotic completeness

We already discussed that Ω± will not in general be unitary as mapsH → H. At finite times t the operatorU(t)U0(−t) is unitary. We know that bound states (any vector from the point-spectrum) cannot appearin the range of Ω± (those states, according to RAGE, would never get out of some finite range).

Although the Ω± will not be unitary as maps on H, they can still be bijections. We would likethem to be bijections between the space of all free states, i.e. all wave packets formed in Pac(H0)Hand all scattering states, which we define as wave packets from Pac(H)H. Note that for H0 = −∆:Pac(H0)H = H. If the Ω± are such bijections, we say that scattering is asymptotically complete.

72

The physical meaning of asymptotic completeness is that scattering states and localized states do notmix and that the dynamics of the unbound (ac) states of the the two systems are equivalent: localizedremains localized, scattering remains scattering. It just does not happen, that a wave packet goes in andturns into a bound state or gets stuck in some other way. All that goes in comes out again. Neitherdoes it happen that a localized state decides to decay. Of course, in physics such processes do seem tooccur, they go by the name of “capture” or “decay” or so: looking more closely, in these processes eitheranother constituent goes out (e.g. a photon is emitted) or some long-lived intermediate state is formed(a resonance), that will decay eventually.

As often, an example where the scattering operators exist, but are not asymptotically complete isinteresting, (see Reed and Simon III, counterexample to Theorem XI.32, p.70).

13.7.2 Scattering operators compare two dynamics

We see that the roles of H0 and H enter symmetrically: the scattering operators connect two unitarilyequivalent dynamical systems. Scattering theory compares one known dynamics to another one, maybeless well known. From a mathematical point of view, it is irrelevant, which is which. Physically, of course,we know everything about, say, −∆ and often rather little about −∆ + V .

But, for example, we know also a lot about the Coulomb problem (all eigenfunctions and generalizedscattering eigenfunctions) and we could just as well use the Coulomb Hamiltonian as our H0 = −∆ −1/r and compare another dynamics, that only asymptotically behaves Coulomb-like. Of course, the ac“eigenfunctions” of −∆ are just plane waves, which are much easier to handle than the hypergeometricscattering functions of the Coulomb problem.

Unbound Coulomb dynamics itself is not unitarily equivalent to free motion. We will need to constructa different dynamics to establish such an equivalence (Dollard Hamiltonian).

13.7.3 Intertwining property

Looking atlim

t→±∞eitHe−itH0e−isH0 = lim

t±∞e−isHeitHe−itH0 (214)

we seee−itHΩ± = Ω±e

−itH0 (215)

Also, the domains are trivially invariant under e−itH0 and ran(Ω±) are invariant under e−itH . As aconsequence, also their complements are invariant under the respective time evolutions: D(Ω±) andran(Ω±) reduce the operators e−itH0 and e−itH respectively. Taking the time derivative at t = 0 we findthe

Theorem 16. (Intertwining property) The subspaces D(Ω±) and ran(Ω±) reduce H0 and H, respectively,and the operators on these spaces are unitarily equivalent:

ΩωH0ψ = HΩ± for ψ ∈ D(Ω±) ∩ D(H0). (216)

With the above we seem to have achieved our goal. However, the domains D(Ω±) may not be all ofP (ac)H, they may even be trivially 0.

First we single out a situation where we obtain results as desired (“Cook’s criterion”):

Lemma 13. (Cook) Suppose D(H0) ⊆ D(H). If∫ ∞0

||(H −H0) exp(∓itH0)ψ|| <∞ for ψ ∈ D(H0) (217)

then ψ ∈ D(Ω±), respectively. It then also holds that

||(Ω± − 1)ψ|| ≤∫ ∞

0

||(H −H0) exp(∓itH0)ψ||dt. (218)

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Proof. We use the Dyson equation

U0(t) = U(t) + i

∫ t

0

U(t− s)(H −H0)U0(s), (219)

or

eitHe−itH0ψ = ψ + i

∫ t

0

eisH(H −H0)e−isH0ψds ∀ψ ∈ D(H0), (220)

and the norm of this is finite by the assumption of the lemma.

With this we can immediately single out a class of potentials where the Møller operators are definedon all H

Theorem 17. Let H0 = −∆ and H = H0 + V where V ∈ L2(R3). Then the Møller operators exist andD(Ω±) = H.

Proof. • We need to show finiteness of

||V ψ(s)|| =∫R|V (x)ψ(s, x)|2dx ψ(s) := eisH0ψ.

Considering ψ like a multiplication operator applied to V , we see

||V ψ(s)|| ≤ ||V ||||ψ||∞ ≤1

(4πs)3/2||ψ(0)||1||V ||, s > 0,

The last inequality will be left for an exercise. This estimate is useful if ||ψ||1 < ∞. with that weget the estimate for the integral∫ ∞

1

||V (ψ(s)|| < 1

(4π)3/2||ψ(0)||1||V ||

Therefore we know ψ ∈ D(H0) and ||ψ||1 <∞ is ∈ D(Ω+). Same procedure for Ω−.

Problem 13.50: Dyson equation Assume that

id

dtψ(t) = H0(t)ψ(t), i

d

dtφ(t) = H(t)φ(t) (221)

define unitary time-evolutions ψ(t) = U0(t, t0)ψ(t0) and φ(t) = U(t, t0)φ(t0). Convince yourself of theidentity (“Dyson equation”)

U(t, t0) = U0(t, t0)− i∫ t

t0

U0(t, s)[H(s)−H0(s)]U(s, t0)ds,

13.8 Asymptotic completeness - guide through the proof by Enss

We have found a case where the Møller operators are defined as we want on D(Ω±) = P (ac)(H0)H. Whatwe do not know yet is, whether the desired property ran(Ω±) = P (ac)(H)H holds. This is the core of theproof of asymptotic completeness. We will follow an elegant proof by Volker Enß[Comm. Math. Phys.61, 285, (1978)]. First outline of the ideas and steps of the proof (following Teschl)

Watch out: recent mathematical physics uses different sign conventions for the scattering operatorsthan physicists. The rift in sign conventions goes right between Reed and Simon, and, for example,Teschl. As we mostly rely on Teschl for the proof, we use this convention.

The Theorem states:

Theorem 18. Suppose V is short range. Then the Møller operators exist and are asymptotically complete.

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13.8.1 Idea of the proof

The idea of Enß’s proof is clearly spelled out in his original paper: take any wave packet ψ ∈ Pac(H) toa remote time in the future e−itHψ = ψ(t). At that time, decompose it into parts that go out, i.e. partswhere position vector and momentum point to the same direction from the scattering potential (assumedto be located near the origin) ~x · ~p > 0, and parts that go in ~x · ~p < 0.

As a first step, this decomposition can be used to show that the scattering operators exist on allP (ac)(H0)H = H via Cook’s criterion. Once we know the scattering operators exist, we also know that(intertwining property)

Ω±Pac(H0)H = Pac(H)Ω±H ⊂ Pac(H)H (222)

As asymptotic completeness is equivalent to

Ω±Pac(H0)H = PacH, (223)

one only needs to showΩ±Pac(H0)H ⊃ PacH, (224)

i.e. any scattering wave packet ψ ∈ Pac(H)H is also ψ ∈ Ω±Pac(H0)H.This second and decisive step is made as follows: at large positive times t, outgoing components

cannot interact in the future, as they will never get near to the potential, from which one concludes thatoutgoing components remain in ran (Ω+). Ingoing components, on the other hand, at sufficiently largepositive times, due to the particular short range nature of the potential, are essentially restricted to afinite-dimensional space. The fact that the corresponding map will be a compact operator will guide ourdefinition of what is “short range”. However, we will also have a more constructive definition about thebehavior of the potential at large r.

As the scattering wave-packet converges weakly to 0, at large times it will become orthogonal to thenear finite-dimensional ingoing components. From this one can conclude that the outgoing wave-packetis φ(t → ∞) ∈ ran Ω+. One remembers that ran Ω± is invariant under the time-evolution, so if everφ(t) ∈ ran Ω± for any time, it stays there for all times. The time-limit t→ −∞ shows φ(t) ∈ ran Ω−.

For the reasoning to work one needs to be able to neglect the potential at one point compared tothe kinetic energy parts of the operator, otherwise the identity of “ingoing” and “outgoing” cannot beestablished with sufficient accuracy. This is what restricts the proof (and the present formulation ofscattering theory as a whole), to short range potentials (see above and Teschl Eqs. (12.29) and (12.30)).

13.8.2 Exclusion of the sing spectrum

Asymptotic completeness requires some reasoning in terms of spectral theory to exclude complicationswith the singular continuous spectrum. Here we are helped by the fact that we know that −∆ has purelyac spectrum. If perturbations are added that are not too wild, we will only generate a few bound states,maybe resonances, but no spectrum of sing character. However, we skip this discussion here and assumethere is no sing spectrum.

13.8.3 In- and outgoing spaces

But how to decompose the wave packet into some kind of in- and out parts without resorting to thescattering operators themselves? The division into in- and outgoing parts of the ψ is achieved by con-structing ~x · ~p as a self-adjoint operator projecting onto the positive and negative spectral subspaces.D = 1

2 (~x · ~p+ ~p · ~x), the self-adjoint generator of the (unitary) dilation group

UλΨ(~x) = (eλ)3/2Ψ(eλ~x), Uλ =: eieλD (225)

By Stone theorem, the generator D of a differentiable, one-parameter unitary group (Uλ) is selfadjoint(Teschl (10.9) and (12.18)). Spectral projectors are used to split the space into parts with positive (out)and negative (in) dilation rate, i.e. P± := PD((0,±∞)).

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Problem 13.51: Dilation group and its generator Coordinate stretching by an overallfactor e−λ is defined as

φ(~r)→ φ(e−λ~r). (226)

(a) Adjust the above idea such that that the stratching map Uλφ is unitary.

(b) Show that the Uλ form a unitary, strongly continuous group.

(c) Show that on a suitable dense set ⊂ H the generator Uλ = exp(iλD) is given by

D =i

2(~r · ~∇+ ~∇ · ~r)

13.8.4 Mellin transform

We also will need an instrument to convert D into a multiplication operator, i.e. a spectral representationof D. This is the Mellin transform, which can be motivated by the following considerations: in polarcoordinates, the dilation stretches (or shrinks) all distances from the origin by a factor eλ, λ ∈ R. Sothe radial coordinate is actually multiplied by a number as in eλr. We can consider the logarithm of thisρ = log(r), for which the stretch will appear as a shift by λ. So that means that D generates a shift whenconsidering its action on ρ. If we take the Fourier transform w.r.t. ρ, D will appear as a multiplicationoperator. Designating ~r = rω we have the Mellin transform

M :LRn → L2(R, Sn−1) (227)

ψ(rω) → (Mψ)(λ, ω) =1√2π

∫ ∞0

r−iλψ(rω)rn/2−1dr (228)

The Mellin transform (as a Fourier transform) is unitary and it converts the stretching transforms U(s)and D to multiplications

M−1U(s)M = e−iλs

M−1DM = λ

As the measure on the space is ac, we immediately know that D has pure ac spectrum.Next, we observe Ff(D) = f(−D)F : the dilation operator changes sign under Fourier transform.

This is to be expected, leaving domain questions aside, as FxF−1FpF−1 = −px.We know that there exists a domain such that D is s.a. One can show that D is essentially self-adjoint

on the Schwartz space, i.e. with D(D) = S(Rn).

Problem 13.52: Dilation operator Show that D, D(D) = S(Rn) is essentially self-adjoint.Hint: Map into a the spectral representation and show ker(D∗) = 0.

13.8.5 Schwartz space

We will be interested in functions that decay well (better than any inverse polynomial), but smoothlyenough such that their derivatives also decay rapidly. This is the Schwartz space of functions

S(Rn) = f ∈ C∞(Rn)| supx|xα(∂βf)(x)| <∞, α, β ∈ Nn0. (229)

For α and β we use a multi-index notation. These functions are dense in the corresponding Hilbert space.

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13.8.6 Compact operators, weak convergence

Note the following: a weakly convergent series φn φ is uniformly bounded supn ||φn|| = M < ∞.One might be worried that a sequence, while turning to ever new directions could also grow in normindefinitely. But this cannot happen for Banach spaces, as a consequence of the uniform boundednessprinciple

Theorem 19. (Banach-Steinhaus) Let X be a Banach space and Y some normed vector space. Let Aαbe a family of bounded linear maps. Suppose ||Aαx|| < Cx is bounded for fixed x ∈ X. Then Aα ≤ C isuniformly bounded.

Proofs for this very fundamental theorem can be found in literature, also Teschl. (Note that in oneof the problems we did not use it and rather require the sequence to be bounded.)

We will define short range potentials such that they generate only relatively compact differencesbetween the two dynamics H and H0. The following equivalent characterizations of compact operatorswill be useful there (Teschel 6.9):

Lemma 14. Let K be a linear map on H. The following statements are equivalent

1. K is compact

2. K∗ is compact

3. An bounded linear maps on H, Ans→ A implies AnK → AK

4. ψn ψ implies Kψn → Kψ.

5. ψn bounded implies Kψn has a norm-convergent subsequence.

Proof. We only show 1.,2.⇒ 3.:

• We consider An → 0 without loss of generality.

• As we have assumed An to be bounded and ||Anψ|| → 0, it is unformly bounded ||An|| < M .Therefore we can study the finite rank case

||AnK|| ≤ ||An||||K|| = ||An||||KN ||+ ||An||ε

•

13.8.7 Short range potentials

The discussion is restricted to H0 = −∆ and H = H0 + V . First requirement is that the potential V be“relatively bounded” to −∆, i.e. ||V (−∆ + 1)−1|| < ∞. Note that, as σ(−∆) = R+ is postive, we canuse the a real offset instead of the complex z for the general self-adjoint case.

A relatively bounded perturbation V of the Laplacian −∆ is called short range if∫ ∞0

||V (−∆ + 1)−11R3\Br ||dr <∞ (230)

where Br := x ∈ R3, |x| < r. Relative boundedness guarantees that the integrand is always finite.This is not as strong a constraint as it may seem: remember that also the Coulomb potential is relativelybounded to −∆. What matters is that the remote parts of the potential drop sufficiently fast comparedto the kinetic energy. That this is not the case for the Coulomb potential.

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Problem 13.53: Relative boundedness Show that the Coulumb potential is bounded relativeto −∆. Show that the Coulomb potential is not short range.

Technically, the short range property is mostly exploited in the form of the compactness of

(H − z)−1 − (H0 − z)−1 and f(H)− f(H0) (231)

Lemma 15. If V is short range, then RH(z)−RH0(z) is compact.

For the proof of the lemma we would need a few results on compactness that we have skipped.However, it gives some insight to consider the ideas of the proof

• Denote χr(x) = χx||x|>r(x), the projector onto the remote parts of space

• So, 1− χr projects onto a finite range.

• We use the (1− χr)RH0(z) is compact (without proof). In fact, the product of any two operatorsg(x)f(p) for integrable functions g, f that deay to 0 at large x and p turns out to be compact (Teschl7.21).

• Also, RH(z)V is bounded: that is plausible, as RH0(z) is bounded and H = H0 +V , closely related

to Kato-Rellich. (Proof in Teschl 6.23).

• Now the short range condition enters

limr→∞

RH(z)V (1− χr)RH0(z)→ RH(z)V RH0(z)

• Note the identity

RH(z)V RH0(z) = (H0 + V − z)−1V (H0 − z)−1 = (H0 + V − z)−1 − (H0 − z)−1

Actually, short range implies that for any function that decays at ∞, we have

Lemma 16. Suppose RH(z)−RH0(z) is compact. Then so is f(H)− f(H0) for any f ∈ C∞(R) and

limr→∞

||[f(H)− f(H0)]χr|| = 0. (232)

Proof. Again the proof involves technical results that were skipped (Teschl 6.21). But once we know thatf(H)−f(H0) is compact, we know that χr excludes an arbitrarily large space, i.e. any χrφ goes stronglyto zero. Multiplying by a compact operator does not change that. As the family of maps is bounded andgoes to zeor strongly, it goes to zero uniformly, i.e. in the operator norm.

13.8.8 Escape of the wave packet - Perry’s estimate

Just how successfully the wave-packet escapes the influence of the potential is quantified by Perry’sestimate (Teschl, Lemma 12.5). To specify this, one needs to battle a few technical complications: firstof all, very slow particles escape the potential very slowly. For any estimate, we need to have a leastan ε of velocity. For any ε we can formulate a sufficiently strong estimate. This still provides a denseset of functions to work with. Secondly, one has a similar problem with the division into in and out:when particles are very close to the origin or spiral very slowly away from the origin, the distinction maynot be sharp enough, although velocities may be high. Again, we need to make the estimate startingfrom some finite dilation rate R by using the projector PD((±R,±∞)) rather than P± = PD((0,±∞)).Perry’s estimate then is

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Lemma 17. (Perry’s estimate) Suppose f ∈ C∞0 (R) with supp(f) ⊂ (a2, b2) for some a, b > 0. For everyR ∈ R, N ∈ N there exists a constant C such that

||χx||x|<2a|t|e−itH0f(H0)PD((±R,±∞))|| ≤ C

1 + |t|N,±t ≥ 0. (233)

Before proceding to the proof, let us look at the meaning of the pieces here:

• First of all, we project onto a subspace of the dilatations, setting a lower of bound R on the out-going(or in-going) nature of the functions. In particular, R = 0 (not expansion at all under stretching)and some open surrounding of it are excluded.

• The we make another spectral cut, this time w.r.t. kinetic energy by excluding too slow and, forconvenience, also too fast particles.

• Then the free time-evolution is applied.

• Finally, we project onto a sphere that grows at the “speed” of the slowest particle admitted in thesystem

• This probablility decays with time faster than any power.

Clearly, this is the general behavior we expect if scattering is to make sense.

Proof. • Let ψ ∈ S(Rn). For the manipulations we we move partially into Fourier space, where f(H0)and the time-evolution have a simple form.

• Introduce

ψ(t, x) =e−itH0f(H0)PD((R,∞))ψ(x)

=〈Kt,x,FPD((R,∞))ψ〉=〈Kt,x, PD((−∞,−R))ψ〉 :

The scalar product is a convenient way of writing the back-Fourier transform to point x. We havedefined

Kt,x(p) =1

(2πn/2)ei(tp

2−px)f(p2)∗

i.e. we have lumped together the time-evolution in p-space, the Fourier transform and the kineticenergy constraint, which is manifestly in L2(d(n)p,Rn). We can apply Schwartz inequality

〈PD((−∞,−R))Kt,x, ψ〉2 ≤ ||PD((−∞,−R))Kt,x||2||ψ||2

which reduces the estimate to

||PD((−∞,−R))Kt,x||2 ≤const

(1 + |t|)2Nfor |x| < 2a|t|, t > 0

• For evaluating ||PD((−∞,−R))K|| we use the Mellin transform (remember it is unitary)

||PD((−∞,−R))Kt,x||2 =

∫ −R−∞

dλ

∫Sn−1

d(n−1)ω|(MKt,x)(α, ω)|2

where the integrand is

MKt,x)(α, ω) =1

(2π)(n+1)/2

∫0

∞f eiα(r)

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• This integral has the form of a smooth function f times a (rapidly) oscillating part (for details seeTeschl). For the case, the method of stationary phase provides a rigorous estimate

|MKt,x)(α, ω)| ≤ const

(1 + |λ|+ |t|)N

which leads to the desired bound (for the missing details, consult Teschl).

Problem 13.54: Method of stationary phase Integrals of oscillatory functions are oftenapproximated in physics by this method. The rigorous mathematical formulation of the method comesin the form of a bound as follows: Consider

If,φ(t) =

∫ ∞−∞

f(r)eitφ(r)dr,

where f ∈ C∞c (R) (i e. a compactly supported infinitely many times differentiable function R → C),φ ∈ C∞(R) real-valued. Suppose now |φ′(r)| ≥ 1 for all r ∈ supp(f) (φ′ denotes the derivative). Show:∀N ∈ N, there is a CN ∈ R, s t. |If,φ(t)| ≤ CN t−N .Hint: Start with N = 0. For N > 0 use a change of variables to show that it is sufficient to look atφ(r) = r. Then use partial integration.

Corrolary 2. Suppose that f ∈ C∞0 ((0,∞)) and R ∈ R. The the operator PD((±R,±∞))f(H0) exp(−itH0)coverges strongly to 0 as t→ ∓∞.

This states that, under suitable spectral cuts, at large postive times there remain no ingoing parts inthe any wave packet undergoing free time-evolution, at large negative times, there are no outgoing part.For the simple proof check Teschl.

Problem 13.55: Corrolary to Perry’s estimate Recall Perry’s estimate: Let f ∈ C∞c (R)with supp(f) ⊂ (a2, b2) for some a, b ∈ R. Then for every R ∈ R, N ∈ N there is C ∈ R, s. t.

‖χx:|x|<2|at|e−itH0f(H0)PD((±R,±∞))‖ ≤ C

(1 + |t|)N,

where PD is the projection valued measure associated to the dilation operator D = (xp+ px)/2.Prove the following corollary: Let f as above and R ∈ R. Then

PD((±R,±∞))f(H0)e−itH0s−−−−→

t→±∞0

(Recall, that strong convergence means convergence of every vector the sequence of operators is appliedto.)

Hint: First split ψ into χψ+ (Id−χ)ψ for a useful χ. Second use Perry’s estimate for the part near theorigin and show that the other part can be neglected. Don’t forget, that we are dealing with boundedoperators!

13.8.9 Existence of scattering operators for short range potentials

The existence of scattering operators is shown by Cook’s criterion, where the boundedness of the integralsis shown by splitting the integral into an inner region and an outer region with a time-dependent radiusseparating the two regions. In the outer region, it is finite because of the short range assumption. In theinner region Perry’s estimate can be administered for a dense set of functions of the form where bothspectral constraints discussed above are applied (Teschl 12.32). Somewhat loosely speaking: scattering

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operators exists because the interacting wave packet leaves from near the potential sufficiently rapidly(Perry estimate) and at larger distances not much happens (short range).

The Møller operators exist by Cook’s criterion, if∫ ∞t0

dt||V e±itH0φ|| <∞ (234)

for all functions φ from a dense subset of Pac(H0)H. We pull out a resolvent RH0(−1) to take advantage

of relative boundedness and split the space into an inner part covered by the short range assumption andan outer part as for Perry’s estimate:

||V e±itH0φ|| =||V RH0(−1) exp(∓itH0)(H0 + 1)ψ||=||V RH0(−1)(1− χ2a|t| + χ2a|t|) exp(∓itH0)(H0 + 1)ψ||≤||V RH0(−1)||||(1− χ2a|t| exp(∓itH0)(H0 + 1)ψ||

+ ||V RH0(−1)χ2a|t|||||(H0 + 1)ψ||

In the second term we have exactly the short range assumption: we require that the integral exists for adense set of functions. The first term is a case of Perry’s estimate (note that 1− χb = χx||x|<b), if werestrict H0 energy and D-spectrum of our test functions by ψ = f(H0)PD((±R,±∞))φ.

Such vectors ψ are dense in H and we know the the Møller operators exist:

D(Ω±) = H. (235)

We do not know yet whether ran(Ω±) is P (ac)(H)D(H), but by the intertwining property we know thatH restricted to ran(Ω±) is unitarily equivalent to H0, and therefore the spectrum σ(Hac) ⊇ σ(H0). AsV was relatively compact, the essential spectra of the two operators agree and as we have excluded thesingular spectrum, we know that σ(H)ac = σ(H0) = [0,∞).

Note that we originally have formulated the Perry estimate only for free motion, but relative bound-edness of the potential ensures that there is no substantial difference between interacting and free motion.

13.8.10 Compactness — in- and outgoing parts are almost free wave packets

We define P± := PD((0,±∞)).

Lemma 18. Suppose V is short range. Let f ∈ C∞0 ((0,∞)), then (Ω± − 1)f(H0)P± is compact.

This Lemma is essential for the prove. Stating that (Ω±−1)f(H0)P± is compact shows that classifyingthe components as in- or outgoing by their behavior under dilatations is meaningful: on the separatepieces, the scattering operators differ “little” from 1 (again with the precaution of not including too smallvelocities by applying f(H0)). “Little” here means by less than ε except on a finite dimensional subspace.

Proof. The proof combines several arguments that we have used before. The following few are specificallyinteresting (for the complete proof → Teschl).

• We will use the characterization of compactness as converting a weakly convergent series ψn intostrongly convergent.

• We write, using the dyson equation:

Ω± − I = limt→±∞

eitHe−itH0 − 1 = +i

∫ ∞0

exp(isH)V exp(−isH0)ds

By this we can estimate

||RH(z)(Ω± − 1)f(H0)P±ψn|| = ||∫. . . ||

≤∫ ∞

0

RH(z)V exp(−isH0)f(H0)P±ψn||ds

81

• To exploit compactness of RHV RH0due to short range, we insert, as before 1 = RH0

(1)(H0 + 1)

• Existence of the integral uses the same splitting into “local” and “remote” parts as in the appli-cation of Cook’s criterion and dominated convergence. Convergence to zero is a consequence ofcompactness of RH(z)−RH0(z).

13.8.11 Final steps of the proof

What is left to show is that any scattering wave packet is ψ ∈ ranΩ+ and ψ ∈ ranΩ−. We will do thisby showing that any scattering wave packet ψ⊥ 3 ranΩ− has ||ψ⊥|| = 0

Proof. • Let ψ ∈ Hc(H) = Hac(H) with ψ = f(H)ψ for f ∈ C∞0 ((0,∞). I.e. ensure there are notany too low (and neither too high) energies in the wave packet.

• Denote ψ(t) = exp(−itH)ψ. By RAGE ψ(t)→ 0 for t→ ±∞

• Separate the in- from the out-parts of ψ(t)

φ±(t) = f(H0)P±ψ(t)

with P± := PD((0,±∞)), P+ + P− = 1 (the missing spectral point λ = 0 is zero weight as σ(D) isac). Note that here we use f(H0) for the spectral cut rather than f(H).

• In spite of the different spectral cuts, the wave packet ψ(t) with finite velocities ultimately decom-poses into in- and out-packets as t→ +∞:

e−itHψ =: ψ(t) ∼ ϕ−(t) + ϕ+(t), ϕ±(t) ∈ P±H (236)

Here short range and ensuing compactness of differences between functions of f(H) and f(H0) isessential:

ψ(t) = φ+(t) + φ−(t) + (f(H)− f(H0))ψ(t) :

as ψ(t) 0 by compactness (f(H)− f(H0))ψ(t)→ 0.

• Again by compactness of (1− Ω±)f(H0), we have

limt→±∞

||(Ω± − 1)φ±(t)|| = 0.

i.e. at sufficiently large times are φ±(t) are almost invariant under the scattering operator.

• Now assume we had some ψ from the orthogonal complement ran(Ω±)⊥. Its norm is

||ψ||2 = limt→∞〈ψ(t)|ψ(t)〉

= limt→∞〈ψ(t)|φ+(t) + φ−(t)〉

= limt→∞〈ψ(t)|Ω+φ+(t) + Ω−φ−(t)〉

• By the intertwining property we know that ran(Ω±) and therefore also ran(Ω±)⊥ are invariantunder the time-evolution exp(−itH). So we know that ψ(t) ∈ ran(Ω±)⊥ for all times t.

82

• By that we know that for the limits t→ ±∞ the component from the respective orthogonal rangeran(Ω±)⊥ cannot contribute to the norm of ψ ∈ Hac(H) and only the terms with the oppositesigns Ω± may give non-zero contributions to the norm of ||ψ||2:

||ψ||2 = limt→±∞

〈ψ(t)|Ω∓φ∓(t)〉

= limt→±∞

〈P∓f(H0)Ω∗∓ψ(t)|ψ(t)〉

= limt→±∞

〈P∓f(H0)e−itH0Ω∗∓ψ|ψ(t)〉

where we have used intertwining in the last step.

• The last term goes to zero by the Corrolary of Perry’s estimate if one takes the limit R → 0.(In physics terms that profits from the fact that at sufficiently large time indeed there remain noingoing contributions in an ac wave packet.)

13.9 Coulomb scattering

[Reed and Simon, section XI.9]

The wave-operators for H = −∆− 1/r and H0 = −∆ do not exist

w − limt±∞

eit(−∆−1/r)e−it(−∆) = 0 (237)

Why? Classical scattering: the hyperbolas of the classical 1/r potential do approach straight linessufficiently well. However, any particle on a Coulomb hyperbola will outrun any free particle with thesame asymptotic energy, no matter how far away from the force-center one starts, specifically with thetime-dependence

r(t) = ct+ d log t+O(1). (238)

While the linear time-dependence is exactly what we expect for free motion, the logarithmic dependencecauses trouble: assume that at one point T0 in time free and Coulomb positions are close. Then a thereis always a later time T1 when the two positions differ by an arbitrary distance L. This, in the end, isalso the reason for the weak convergence → 0 of the quantum scattering operators.

How to repair? Choose a different comparison time-evolution! Do not compare to a free particle, butto something that gets accelerated just enough to keep up with the Coulomb particle. Define Dollardwave operators with the time-dependent Hamiltonian

HD(t) = −∆− 1

2|t|√−∆

θ(−4|t|∆− 1) (239)

and it turns outΩ

(D)± = s− lim

t→±eit(−∆−1/r)e−i

∫ t0HD(s)ds (240)

exists. Again Volker Enss (1979) proved asymptotic completeness for Dollard comparison operators (notin the 1972 Reed and Simon book).

Apart from the technical complications, does HD define useful dynamics for the comparison? It turnsout yes, as momentum information that we are interested in is unaltered. The long time limit of momentais the same as for free motion, this is what is usually measured. What changes is the exact interpretationof arrival times at the detector, but that is usually not considered. Actually, it is hard to observe as ina typical scattering experiment there is no information on when exactly the interaction takes place, andtherefore how long time the particle traveled before it hit the detector. (Note the arrival times can bemeasured very accurately.)

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13.9.1 Exact solutions

Like in the classical case, we also know the exact scattering solutions of the quantum Coulomb problem.And we know the mapping of ingoing to outgoing asymptotic momenta. In that sense the single particleCoulomb scattering problem can be considered as solved.

The reason for this solvability is the same as in the classical case: high symmetry of the system.Except for angular momentum and energy, there is one more conserved vector: the Lenz-Runge vector:the “perihelium” position in classical mechanics.

It turns out that “asymptotic momentum operators” p± can be defined (compare the classical hy-perbolas), in terms of which HPac = p2

±. After that, the scattering problem can be solved with purelyalgebraic methods (for details, see Thirring).

13.9.2 Coulomb + finite range potentials

As the Coulomb dynamics is known completely, we can use it instead of the free motion as a comparisontime evolution for dynamics, that are long range Coulomb but have short range modifications. Think ofthe scattering from the ion of a molecule.

Suppose we have a Hamiltonian of the form

H = −∆− 1/r + V = Hc + V (241)

where V (Hc − i)−1 is bounded and short range. Then the same apparatus of scattering theory can berun for the wave operators

Ω(c)± = lim

t→±∞eitHe−itHc (242)

which will map H(Hc)ac → H(H)ac. Existence of the scattering operators is proven (e.g. Muleherin and

Zinnes, 1970). Asymptotic completeness ranΩ(c)± = H(H)ac, can be inferred from the result of Enss the

general asymptotic completeness for long-range (∼ 1/r) potentials relative to Dollard operators.

14 Formal derivations

In this section we do not even try to give rigorous results. Rather, we show a few manipulations thatrelate the rigorous results of the previous section to what physicists often do.

14.1 The Lippmann-Schwinger equation in disguise

The LS equation is of great practical importance in physics. We introduced it as an equation for the

“scattering eigenfunctions” HΨ(±)~k

= k2

2 Ψ(±)~k

. Unfortunately, rigorous foundation of “scattering eigen-functions” and derivation of related formulae of practical importance requires the buildup of a significantmathematical apparatus. Here we will introduce the “dirty” physicist’s formal manipulations that leadfrom the scattering operators to the LS equation and further on to cross-sections.

Take an arbitrarily narrow wave free wave packet “ϕ~k ∼ ei~k·~r”, then

|Ψ(±)〉 := Ω±|ϕ~k〉 (243)

is indeed a seemingly much simpler equation than the LS equation to obtain scattering states Ψ(±). TheΨ(+) is what we want as an “in” state: in the far past it looks like the free wave packet ψ~k. The somewhatawkward sign convention where the far past corresponds to + has historic reasons. It was not originallychosen for the time limits. Rather, in the LS equation, the sign k2/2+ iε corresponds to complex energiesin the imaginary upper half-plane, which is the part relevant for the time-limit to the past.

The similarity becomes even more evocative, when we will show that in the spectral representationof H0 we have

Ω± = 1− lime↓ε

∫ ∞0

(H − E ∓ ıε)−1V δ(H0 − E)dE (244)

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We use the notation for the spectral measure dP (E) = δ(H0 − E)dE (we have a pure ac spectrum andtherefor no trouble with possible point spectrum. Some signs need adjustment, but most importantly wehave the resolvent of H, not H0. Unfortunately, this makes this formula useless for computations. Butit can be transformed to the LS equation.

14.2 Ω± in the spectral representation of H0

Ω± are time-independent operators operators obtained as time limits of time-dependent operators. Herewe will replace the limit in time by a limit of resolvents of H0. This is achieved by using a result by Abel:

Let f be a bounded function on (0,∞) and suppose limt→∞ f(t) = f∞. Then

limε↓0

ε

∫ ∞0

e−εtf(t)dt = limt→∞

f(t). (245)

Problem 14.56: Abel limit Show that

V (t)→ V∞ ⇒ limε↓0

ε

∫ ∞0

e−tεV (t)dt = V∞

as a strong limit. In the same way we can evaluate the limits of eitHe−itH0 , where we heavily rely on theintuition that we can treat operators just like numbers as long as we respect their non-commuting nature(and take care of domain questions).

In the spectral representation of H0

H0 =

∫ ∞0

EdP (E) (246)

we write

eitHe−itH0 = eitH∫ ∞

0

e−itEdP (E) =

∫ ∞0

eit(H−E)dP (E) (247)

The integrals exists, as the involved functions and operators are bounded. Now we use the Abel methodand formally the time-integral

Ω± = limε↓0

ε

∫ ∞0

e−εt∫ ∞

0

e±it(H−E)dP (E)dt (248)

= limε↓0

ε

∫ ∞0

∫ ∞0

e±it(H−E±iε)dt dP (E) (249)

= limε↓0±iε

∫ ∞0

(H − E ± iε)−1dP (E) (250)

= 1− limε↓0

∫ ∞0

(H − E ± iε)−1V dP (E) (251)

The last step is by some algebra with the resolvent

(H − E ± iε)−1(±iε) = (H − E ± iε)−1[(H − E ± iε)− (H − E)]

= 1− (H − E ± iε)−1(H0 + V − E) (252)

and observing that(H0 − E)dP (E) = (H0 − E)δ(H0 − E)dE = 0.

For the time integral we have treated H like an ordinary function: as it is selfadjoint, it has a spectralrepresentation

H =

∫GdPH(G) (253)

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for the projection valued measure PH belonging to H. Then∫ ∞0

e±it(H−E±iε)dt =

∫ ∫ ∞0

e±it(G−E±iε)dP (G)dt (254)

with G ∈ σ(H) ⊂ R. Integrals can be interchanged, the time integral can be performed and the spectralrepresentation can be replaced by the abstract operator again.

Thus we arrive at

Ψ(±) = Ω±φ = φ− limε↓0

∫ ∞0

(H − E ± iε)−1V dP (E)φ, (255)

which bears great similarity with the LS equation. Actually, it is seemingly better, as it is a directexpression for some scattering wave packet Ψ(±) rather than an integral equation for it. However, itcontains (H − E ± iε)−1, which usually is rather inaccessible to computation.

Now we remember the symmetric role of H and H0 in scattering theory and we interchange them

H,H0, H −H0 =: V → H ′ = H0, H′0 = H,H ′ −H ′0 = H0 −H = −V (256)

to write

φ(±) = ψ + limε↓0

∫ ∞0

(H0 − E ± iε)−1V dPH(E)ψ, ψ ∈ Pac(H)H. (257)

Next remember that the signs ± just indicate whether the φ(±) and ψ are mapped by going through thethe remote past or future. That is, we can just as well move ± from φ back to ψ, leading to

ψ(±) = φ− limε↓0

∫ ∞0

(H0 − E ± iε)−1V dPH(E)ψ(±). (258)

14.2.1 Remark on “ac-spectral eigenfunctions”

The equation above is almost the shape of the LS equation. However, it is formulated for wave-packetsrather than, as physicists like to do it, for “eigenfunctions of the continuous spectrum”. For −∆ thesewould be plane waves exp(i~k · ~x). Indeed, the path for making this notion precise, is through the Fouriertransform and using the mapping by Ω±, formally

|~k〉(±)H = Ω±|~k〉H0

:= Ω±ei~k·~x. (259)

The mapping is not unique because of the two alternative signs, but it does not need to be, as the energiesare expected to be degenerate. Time-reversal maps between the two signs. Accepting that, we can evengive meaning to the δ-function of the Hamiltonian as the spectral projector onto the eigenfunctions ofgiven energy ∫ ∞

0

δ(H − E)dE :=

∫dαdk2 k

2|k, α〉〈k, α| (260)

=

∫ ∞0

dE

∫dαE1/2

2|√E,α〉〈

√E,α| (261)

=

∫ ∞0

dE

∫dα|E,α〉〈E,α| (262)

where

|E,α〉 =E1/4

√2|√E,α〉. (263)

If we do that, formally, the δ(H−E) allows to perform the dE integration and we have the Lippmann-Schwinger equation for the scattering solutions

ψ(±)(E) = φ(E)− limε↓0

(H0 − E ± iε)−1V ψ(±)(E). (264)

where the + corresponds to matching of asymptotics of φ = ei~k~r in the remote future (“outgoing boundary

conditions”) and − to matching in the remote past (“ingoing boundary conditions”).

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14.2.2 The LS equation in position space

In position space, the LS equation takes the form

ψ(±)(~r) = ei~k~r −

∫R3

dr(3) e∓ik|~r−~r′|

4π|r − r′|V (~r)ψ(±)(~r′). (265)

where we recognize the form of the solution ψ(±)(~r) as a plane wave plus a superposition of sphericalwaves.

Problem 14.57: Green’s function in position space Compute the function

G±(~r, ~r′) =1

(2π)3

∫d(3)k

ei~k(~r−~r′)

k20 − k2 ± iε

= − 1

4π

e±ik0|~r−~r′|

|~r − ~r′|

(We use the physicists’ convention not to explicitly write the limε).Hint: Use polar coordinates, expansion of a plane wave into spherical Bessel functions, integrate overthe angle, use the residue theorem for what remains.

14.3 Mapping free to free spaces: the S-matrix

With asymptotic completeness the Ω± unitary bijections Pac(H)H ↔ Pac(H0)H and the S-matrix mapsPac(H0)H → Pac(H0)H

S = (Ω+)−1Ω− = Ω∗+Ω− (266)

This corresponds to a limit

limt→∞

[U(−t)U0(t)]−1[U(t)U0(−t)] = limt→∞

U0(−t)U(2t)U0(−t)] (267)

i.e. bring a free packet sufficiently far into the past, let it interact for sufficiently long time (into thefuture), propagate the free packet it back into the present: generate the effect of scattering as a mappingbetween free wave packets.

Some properties of the S-matrix:

• is unitary (asymptotic completeness)

• it “inherits” symmetries from H and H0.

• it conserves energy - intertwining relation

14.3.1 Spectral representation of the S-matrix

In the same spirit as for Ω±, there is a representation of the S-matrix in the spectral representation ofH0:

S = limε↓0

∫ ∞0

dE

1− 2πıδ(H0 − E)[V − V (H − E − ıε)−1V

]δ(H0 − E). (268)

We use the (only slightly) symbolic notation

δ(H0 − E) =∑∫

α

|E,α〉〈E,α|, (269)

where α labels the degeneracy in the energy subspace E.Energy conservation follows from the intertwining property of the Møller operators

SH0 = Ω−1+ Ω−H0 = Ω−1

+ HΩ− = H0Ω−1+ Ω− = H0S.

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It implies that S is a function of H0, i.e. in the spectral representation is a multiplication by a functionof E. It may still connect different α, though (after all, we are free to choose the basis in the space offixed E spanned by the |E,α〉. In the rotationally symmetric case, S =

⊕lm Sl we know that l,m is

not changed and therefore can be parameterized by the “scattering phases” σl(k), conventionally writtenwith k =

√E:

S|E, l,m〉 = e2iδl(k)|E, l,m〉 (270)

Note that for rotationally symmetric problems there cannot be any dependence on m.The label α can be discrete or continuous. Think of polar angles to denote directions, or, alternatively,

the corresponding resolution of the rotations in terms of sphereical harmonics:

|α〉〈α| = |φ, θ〉〈φ, θ| = P (φ, θ) =∑lm

Ylm(φ, θ)Y ∗lm(φ′, θ′) = δ(θ − θ′)δ(φ− φ′) (271)

14.3.2 The T -matrix

Let ψ,ψ be two wave packets. The matrix elements of the S-matrix can be written as

〈ψ, Sψ〉 = 〈ψ,ψ〉 − 2iπ

∫d(3)k ψ(~k)∗T (~k,~k′)δ(~k2 − ~k′2)ψ(~k′) (272)

where ψ(~k) is the representation of ψ in ~k-space with the T-matrix

T (~k,~k′) = 〈~k|V − V (H − E − ıε)−1V |~k′〉, (273)

where |~k〉 = exp(i~k~x)/(2π)3/2 are the δ-normialized plane waves 〈~k|~k′〉 = δ(3)(~k − ~k′) and ψ(k) = 〈~k|ψ〉.T (~k,~k′) contains the relevant scattering information. Also, it is a smooth function of ~k,~k′, loosely

speaking, where solutions of the LS equation exist. It essentially is the same as the scattering amplitudef(k, ~n, ~n′)

f(k, ~n, ~n′) = −(2π)2T (k~n, k~n′) (274)

whose square gives the differential cross section:

σ(~k,~k′) = |f(k, ~n, ~n′)|2. (275)

14.4 Scattering cross section — derivation

Notation In the following, for easier connecting to standard literature, we assume mass =1, i.e. H0 =−∆/2 and the E = k2/2.

14.4.1 Post and prior form

Note that on the level of generalized eigenfunctions/the LS equation we have

|ψ(±)(~k)〉 = Ω±φ~k =

[1− lim

ε↓0(H − k2 ± iε)−1V

]|φ~k〉 (276)

and we recover the known formulae of scattering theory

− 1

2π2f(~k,~k′) = 〈φ~k|V − V (H − k2 + iε)−1V |φ~k′〉 (277)

= 〈φ~k|V Ω+|φ~k′〉 = 〈φ~k|V |ψ(+)~k′〉 (278)

= 〈Ω−φ~k|V φ~k′〉 = 〈ψ(−)~k|V |φ~k′〉 (279)

These are the post(+) and prior(-) form, respectively, of the scattering amplitude. The two form arerigorously equal. In practice the distinction can become important in a multi-channel situation, where

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the left two scattering solutions ψ(+) and ψ(−) differ and approximations to one of them can be moreappropriate than to the other.

Problem 14.58: First Born approximation Compute the scattering amplitude for theYukawa potential

V (~x) = V0e−µ|~x|

µ|~x|, µ > 0 (280)

in “first Born approximation”, expressing ψ(±) by its the first term of the Born series. Express results asa function of cos γ, where cos γ = ~k~x/(|~k||~x|), γ the angle between ~k and ~x.

(a) How does the first Born cross section depend on the sign of V0?

(b) How does the cross section behave with |~k| → 0?

(c) What is the total cross section ∫ 1

−1

d cos θ

∫dφ|f (1)(~k,~k0)|2? (281)

θ and φ are the polar angles of ~k = |~k0|(sin θ cosφ, sin θ sinφ, cos θ)

Hint: Use the expansion of a plane wave into its spherical components

eı~k~x =

∑l

(2l + 1)Pl(cos γ)ıljl(|~k||~x|) (282)

where the Pl are Legendre polynomials.

P0 = 1, P1(x) = x, P2(x) =1

2(3x2 − 1) . . . (283)

with the spherical Bessel functions

j0(x) =sin(x)

x(284)

j1(x) =sin(x)

x2− cos(x)

x(285)

j2(x) =

(3

x2− 1

)sin(x)

x− 3 cos(x)

x2(286)

. . . (287)

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