Fractal Weyl law for three-dimensional chaotic hard-sphere ... · Fractal Weyl law for...

Fractal Weyl law forthree-dimensional

chaotic hard-sphere scatteringsystems

Diplomarbeitvon

Alexander Eberspächer

31. März 2010

Hauptberichter: Prof. Dr. Jörg MainMitberichter: Prof. Dr. Günter Mahler

1. Institut für Theoretische PhysikUniversität Stuttgart

mailto:[email protected]

http://itp1.uni-stuttgart.de

http://www.uni-stuttgart.de

Contents

1 Introduction 1

2 The fractal Weyl law 32.1 The Weyl law for closed systems . . . . . . . . . . . . . . . . . . . . . . . 32.2 The repeller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 The fractal Weyl law for chaotic open systems . . . . . . . . . . . . . . . 5

3 Gauging the repeller 93.1 Introducing the four-sphere billiard . . . . . . . . . . . . . . . . . . . . . 93.2 Playing billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Fractal repeller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3.1 Time-delay functions . . . . . . . . . . . . . . . . . . . . . . . . . 123.4 Hausdorff dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4.1 Estimating dH through Hausdorff sums . . . . . . . . . . . . . . . 163.5 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.5.1 Interval bisection in billiards . . . . . . . . . . . . . . . . . . . . . 183.5.2 Finding regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.5.3 Building Hausdorff sums . . . . . . . . . . . . . . . . . . . . . . . 21

3.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Semiclassical quantisation of quantum billiards 274.1 From the trace formula to zeta functions . . . . . . . . . . . . . . . . . . 27

4.1.1 The trace formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1.2 The trace formula for billiards . . . . . . . . . . . . . . . . . . . . 294.1.3 The Gutzwiller-Voros zeta function . . . . . . . . . . . . . . . . . 31

4.2 Symbolic dynamics and periodic orbits in billiards . . . . . . . . . . . . . 324.2.1 Full symbolic codes . . . . . . . . . . . . . . . . . . . . . . . . . . 324.2.2 Symmetry reduction . . . . . . . . . . . . . . . . . . . . . . . . . 334.2.3 Finding periodic orbits . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3 Cycle expansion of zeta functions . . . . . . . . . . . . . . . . . . . . . . 354.4 Discrete symmetries and cycle expansion . . . . . . . . . . . . . . . . . . 39

4.4.1 The origin of degeneracy from symmetry . . . . . . . . . . . . . . 394.4.2 The symmetry group Td . . . . . . . . . . . . . . . . . . . . . . . 414.4.3 Using symmetries in cycle expansions . . . . . . . . . . . . . . . . 42

iii

Contents

4.5 Harmonic inversion method . . . . . . . . . . . . . . . . . . . . . . . . . 494.5.1 Harmonic inversion technique . . . . . . . . . . . . . . . . . . . . 494.5.2 Investigating regions deeper in the complex plane . . . . . . . . . 554.5.3 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Results 595.1 Defining a scale for the strip widths . . . . . . . . . . . . . . . . . . . . . 59

5.1.1 Calculating the escape rate . . . . . . . . . . . . . . . . . . . . . . 605.2 Resonance spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.2.1 Comparing different orders of cycle expansion in different subspaces 615.2.2 Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.3 Counting resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.4 Putting the fractal Weyl law to test . . . . . . . . . . . . . . . . . . . . . 73

5.4.1 Configuration d/R = 6 . . . . . . . . . . . . . . . . . . . . . . . . 735.4.2 Configuration d/R = 8 . . . . . . . . . . . . . . . . . . . . . . . . 745.4.3 Configuration d/R = 10 . . . . . . . . . . . . . . . . . . . . . . . 74

5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6 Conclusion and outlook 79

Bibliography 81

iv

1 Introduction

The asymptotic eigenvalue distribution of partial differential equations such as theSchrödinger equation for free particles or the one-dimensional Helmholtz equation forsound waves has been of interest as early as from 1912 when Hermann Weyl and RichardCourant first studied the problem [1, 2]. Weyl and Courant found expressions for theasymptotic level number in closed systems. The so called “Weyl law”, which has beenwell known from then on, states that for closed quantum systems, every available Planckcell in phase space is occupied by one quantum state. The Weyl law establishes a relationbetween the dimension of phase space and the number of eigenvalues below a certainvalue for bound systems as a simple power law. However, a version of Weyl’s law foropen systems has not been established until the early 2000s [3].

With the so called fractal Weyl law [3, 4] it is conjectured that for open systems arelation similar to Weyl’s law holds. However, for the fractal Weyl, the dimension of thewhole phase space volume available to closed system is conjectured to be replaced by thedimension of the so called repeller, i.e. the set of trapped trajectories. This set is generallya fractal, so the dimension entering into the fractal Weyl law can be a non-integer fractaldimension. Thus, just as in Weyl’s law for closed systems, a classical object, namely therepeller, is connected to a purely quantum mechanical quantity, namely the asymptoticeigenvalue number.

In all publications so far, the conjectured fractal Weyl law was supported by numericalevidence. However, so far, only two-dimensional systems have been investigated. Nu-merical studies in three spatial dimensions or a rigorous proof have not yet been published.

In this thesis, the fractal Weyl law will be put to test for the three-dimensional four-sphere scattering system. Classically, the scattering is chaotic in this system. Firsttheoretical and experimental investigations of the four-sphere scattering system havealready been published by Sweet et al. and Motter et al. [5, 6, 7]. However, theconfigurations studied so far have been limited to a narrow interval of centre-to-centreseparations. In this thesis, a method to compute the fractal dimension of the classicalrepeller for a wide range of the configuration parameter will be presented.

The other quantity needed for an investigation of Weyl’s problem is the number ofquantum resonances for the system. The four-sphere scattering system has been studiedquantum mechanically before [8], but, however, the energy range studied was not sufficient

1

1 Introduction

to put the fractal Weyl law to test. In this thesis, semiclassical approximations will bediscussed and applied to the four-sphere billiard. For the first time, the symmetries ofthis system will be taken into account and the fractal Weyl law will be tested in varioussymmetry subspaces of the problem.

This thesis is outlined as follows: In chapter 2, the fractal Weyl law will be stated anddiscussed. Relevant notions will be defined. Chapter 3 will introduce a novel methodto calculate the fractal dimension of the repeller; whereas in chapter 4, methods ofsemiclassical quantisation of the four-sphere billiard will be presented. The method ofcycle expansion will be discussed in details. The symmetry decomposition of chaoticdynamics and furthermore the high-resolution spectral analysis by harmonic inversionwill be presented. Finally, in chapter 5, results of our investigations will be given anddiscussed.

2

2 The fractal Weyl law

In this chapter, Weyl laws are introduced as statements about the asymptotic distributionof eigenvalues for scalar hyperbolic partial differential equations such as the Schrödingerequation for free particles or the Helmholtz equation for sound waves. In this chapter,both closed and open systems will be discussed. Then the fractal Weyl law for opensystems will be stated and basic definitions of relevant notions for the fractal Weyl lawwill be given.

2.1 The Weyl law for closed systems

H. Weyl and R. Courant studied the asymptotic eigenvalue distribution for closed systemsdefined on a finite spatial region G with either Dirichlet (vanishing wave function ψ|∂G = 0on the boundary) or Neumann (vanishing normal derivative ∂nψ|∂G = 0 on the boundary)boundary conditions [1, 2]. The derivation for closed billiard systems with d spatialdegrees of freedom may be carried out easily.

First we consider closed quantum systems defined on a given volume V . Let the systembe described by the linear partial differential equation

[∆ + k2

]ψ(r) = 0 . (2.1)

For the quantum wavefunction ψ(r), the eigenvalues k may be written as

k2 =2mE

~2, (2.2)

which is a nonlinear dispersion relation. Henceforth, the wavenumber k will loosely becalled “energy”. The number of eigenvalues N(k) up to a given energy k may be calculatedin the 2d-dimensional phase space. The level number N(k) is the energy sphere’s volumedivided by the volume occupied by one single state. In billiards, particles are free except

3


for the encounters with the scattering centres. In such systems, the phase space volumeper state is exactly one Planck cell V0 = (2π~)d [9]. We then write

N(k) =1

(2π~)d

∫ddp

∫ddqΘ(E −H(p))

=V

(2π)d

k∫0

kd−1 dk

∫∂V

dd−1Ω

=V kd

Γ(d/2)(4π)d/2∝ kd . (2.3)

The volume V enters into the result through spatial integration. The integration overthe d-dimensional momentum sphere yields the other terms. The result N(k) ∝ kd hasbeen well-known [10].Later in this chapter, the fractal Weyl law will be introduced as a generalisation of

the Weyl law. The fractal Weyl law is conjectured to hold for open scattering systems,where, instead of the integer dimension of space d, the (possibly non-integer) Hausdorffdimension dH of the so called repeller enters into the asymptotic level number N(k).

2.2 The repeller

Preparing for stating the fractal Weyl law, we first have to introduce the notion of therepeller. The repeller is the set of phase space points given by trajectories q,p that staytrapped for either t→∞ or t→ −∞. Thus, the repeller is a purely classical quantity.Aiming at a more mathematical definition, we may follow [3] and define the repellerKE for the energy E as

KE = K ∩ H = E , (2.4)

where

K =

q,p : sup

t

∣∣∣∣Φt(q,p)∣∣∣∣ <∞ (2.5)

is the set of trapped trajectories. The symbol Φt denotes the Hamiltonian flow.For scattering systems with chaotic dynamics the repeller typically is a fractal object in

phase space. The trajectories do not fill the phase space densely; the repeller may ratherbe compared to a sponge. A characteristic quantity for the sponge’s “bulkiness” is givenby the fractal dimension (also called Hausdorff dimension) which will be introduced insection 3.4.The repeller may also be described in terms of stable manifolds Ws and unstable

manifolds Wu for the trajectories. Denote the phase space for a system with d spatial

4

2.3 The fractal Weyl law for chaotic open systems

degrees of freedom byM ⊆ R2d. Then, the stable manifold Ws(x0) for a phase spacepoint x0 = (q0,p0) is defined as

Ws(x0) =x ∈M :

∣∣∣∣Φtx−Φtx0

∣∣∣∣→ 0 for t→ +∞. (2.6)

Reversing the time direction, we obtain a similar definition for the unstable manifold Wu:

Wu(x0) =x ∈M :

∣∣∣∣Φtx−Φtx0

∣∣∣∣→ 0 for t→ −∞

(2.7)

As the union of all stable or unstable manifolds of all points on a given trajectory isinvariant under time evolution, it is called the invariant manifold. Now, the repeller maybe understood as the union of all invariant manifolds of all trajectories of the system forfixed energy E. The periodic orbits which will be used in the semiclassical quantisationof the system are a subset of the repeller.


The asymptotic eigenvalue distribution is also of interest for open systems. In thesesystems, the dynamics is not bound, i.e. particles are allowed to escape to infinity. Suchsystems may be realised by microwave or optical cavities. Depending on the shape of thescatterers, the dynamics may either be regular or chaotic. In this thesis convex-shapedscatterers will be considered. The dynamics is thus defocussing as illustrated in figure 2.1.Neighbouring initial conditions lead to exponentially separating trajectories, thus thedynamics is chaotic. In this case, the trapped set – the repeller – is a fractal object ofnon-integer Hausdorff dimension dH.

In the quantum mechanical treatment of scattering systems without bound states, theHamiltonian H is no longer an Hermitian operator, as the wavefunctions ψ(r) do nolonger decay to 0 as r →∞. As a consequence, one way of describing the system is bygoing from real energies E to complex resonances E = E0 − iΓ/2 [11]. The imaginarypart Im(E) = −Γ/2 is related to a state’s lifetime. This can be seen heuristically if weconsider the time evolution of a state:

|ψ(t)〉 = e−iHt/~ |ψ(0)〉= e−iE0t/~ e−Γt/2~ |ψ(0)〉 . (2.8)

Calculating a state’s norm 〈ψ|ψ〉 explicitly, we understand how a negative imaginarypart ImE leads to the state’s decay:

〈ψ(t)|ψ(t)〉 =⟨ψ(0)

∣∣ ei(E0+iΓ/2)t/~ e−i(E0−iΓ/2)t/~∣∣ψ(0)⟩

= e−Γt/~ 〈ψ(0)|ψ(0)〉︸︷︷︸≡1

. (2.9)

5


−4 −2 0 2 4 6x

−4

−2

0

2

4

6

y

Trajectories

Discs

Figure 2.1: Illustration of defocussing dynamics in the three disk billiard. A bundleof 80 initial conditions (red lines starting around the origin) is iterated. As canbe clearly seen, after only a few reflections at the scatterers (drawn in green), theindividual trajectories have separated dramatically. The same behaviour is as wellencountered in the four-sphere scattering system.

This statement holds qualitatively as well for the imaginary part of the wavenumberk =√

2mE/~.Aiming at a generalisation of Weyl’s law to open systems, it has carefully to be defined

how states are counted in the complex plane. The level number N(k) we will use fromnow on for an open scattering system with energies kn is defined as

N(k) = kn : Re(kn) ≤ k; Im(kn) > −C . (2.10)

Typically, there are infinitely many scattering resonances located in the lower half-planeeven for finite Re(k). Fast-decaying states are removed from counting by the cutoffconstant C. This way of counting is illustrated in figure 2.2.The fractal Weyl law was conjectured based on the works of Sjöstrand and Zworski

[12, 13]. The asymptotic level number N(k) is written as a power law

N(k) ∝ kα , (2.11)

where it is conjectured that the exponent

α =D + 1

2(2.12)

6


Re k →

Imk→

Im k=−C

Im k=0

countednot counted

Figure 2.2: Illustrating the counting of resonances in the complex plane. Only resonanceslocated in the strip 0 ≤ Im(k) ≤ −C (red crosses) are counted for N(k). For studyingthe fractal Weyl law, C, therefore, may be considered as a free parameter as it isconjectured that the fractal Weyl law (2.15) holds for all strip widths C.

depends on the fractal dimension D = dim(KE) of the repeller. This exponent is supposedto be valid for dynamics obeying the energy condition H = E. Calculating the fractaldimension in a suitable Poincaré surface of section (PSOS), the Hausdorff dimensiond = D − 1 is diminished by 1 [14]. The fractal Weyl law then reads

N(k) ∝ kd/2+1 . (2.13)

For systems with time reversal symmetry, we will not consider sets of trajectories thatare trapped for both t → −∞ and t → +∞ individually, but rather only the stablemanifold Ws that stays trapped for positive time direction t→ +∞. As the Hamiltoniandynamics in this system is invariant under time reversal T : t→ −t, the stable manifoldWs and the unstable manifold Wu will have the same Hausdorff dimension dH = d/2. Wethus may write

α =d

2+ 1

= dH + 1 , (2.14)

7


which leads to the fractal Weyl law in the form in which it will be used throughout thisthesis:

N(k) ∝ kdH+1 . (2.15)

The Weyl law for closed systems suggests one state per occupied Planck cell in theavailable phase space. Thus, when the dynamics is trapped, every Planck cell inside theenergy hypersurface is occupied. The fractal Weyl law for open systems replaces thewhole phase space accessible with the repeller. The resonances are located on the repeller[15], and the asymptotic level number is given by N(k) ∝ kdH+1.So far, a rigorous proof for the fractal Weyl law is still missing, however, the fractal

Weyl law was numerically verified for various systems, for example for the 3-disk billiard[4], a microstadium cavity [14], a triple Gaussian potential [3], a modified Hénon-Heilespotential [15] and the kicked free particle [16]. All investigations so far have been carriedout for systems with only two spatial degrees of freedom. A first numerical test of thefractal Weyl law for the four-sphere scattering system’s three spatial dimensions is givenin this thesis.

In conclusion, the fractal Weyl law gives an interesting connection of classical mechanicswith the statistics of a quantum system. The left hand side of the power law (2.15)is a purely quantum mechanical quantity, whereas the right-hand side depends on thedimension of a classical object, namely the repeller.

8

3 Gauging the repeller

As the fractal dimension of the repeller enters into the fractal Weyl law (2.15), it isessential to understand how to find the repeller in phase space and how to gauge it. Amethod to find regions of phase space approximating the repeller will be introduced inthis chapter. Furthermore, the basic notion of the fractal or Hausdorff dimension will beintroduced. Finally, it will be demonstrated how the repeller’s approximation can begauged and results for the four-sphere billiards will be given.

3.1 Introducing the four-sphere billiard

A billiard is a system with one or more infinitely high potential barriers. Depending onthe shape of the boundaries, a billiard may either be open or closed. The system underconsideration in this thesis is the four-sphere billiard. This system is characterised byfour spheres of the same radius R located on the vertices of an equilateral tetrahedronwith edge length d as visualised in figure 3.1. A convenient configuration of the billiardsystem is given by placing spheres of radius R at the positions

(x1, y1, z1

)=

(0, 0,

√2

3

), (3.1a)

(x2, y2, z2

)=

(1

2,− 1

2√

3, 0

), (3.1b)

(x3, y3, z3

)=

(−1

2,− 1

2√

3, 0

), (3.1c)

(x4, y4, z4

)=

(0,

1√3, 0

). (3.1d)

As the centre-to-centre separations are

|ri − rj| = 1 , i 6= j ; (3.2)

this configuration corresponds to spheres on the vertices of a unit tetrahedron. The ratiod/R is invariant under isotropic coordinate scaling

r → γr (3.3)

9


Figure 3.1: The four-sphere billiard visualised. Four spheres of equal radius R, drawnin blue, are located on the vertices of an equilateral tetrahedron (indicated by greybars) with edge length d. Shown is the case of d/R = 6. Even for the case of touchingspheres, d/R = 2, the system is open as particles may escape through the “holes” inbetween the spheres.

by a factor γ. Therefore it is only necessary to fix the ratio d/R, which, in the configurationdescribed by equations (3.1) can be established by a simple change of the radius R,whereas all centre coordinates remain the same. For fixed d = 1, allowed values for theradius R are (0.0; 0.5]. The former describes the case of infinite separation of the spheres,the latter is just the case of touching spheres. The invariance of the fractal dimension dH

under this scaling will be demonstrated in section 3.4.

3.2 Playing billiard

In order to find the parts of the phase space that belong to the repeller, we first need tounderstand how to “play billiard”, i.e. how to iterate given initial conditions classically.In this section, the classical dynamics of billiard systems will be described. We considerf -dimensional billiards in which the scatterers are f -dimensional hyperspheres of radiusR, i.e. disks in f = 2 dimensions and spheres in f = 3 dimensions. The i-th sphericalscatterer is conveniently described by the equation

(r − ri)2 = R2 (3.4)

where the vector ri stands for the i-th sphere’s centre. The collision of a particle in freeflight with a hard scatterer is fully elastic and thus conserves the energy E = p2/2m =mv2/2. Therefore, the velocity’s magnitude v is conserved. To describe the reflection,

10

3.2 Playing billiard

we may decompose the incident velocity v− in components parallel and perpendicular tothe vector n0 pointing from the particle to the scatterer’s centre. Choosing n0 as a unitvector, we have

v− = v−‖ + v−⊥ (3.5)

with

v−‖ =(v− ·n0

)n0 ,

v−⊥ = v− − v−‖ . (3.6)

Reflection changes the sign of the parallel component v−‖ , whereas the perpendicularcomponent remains unchanged. The velocity after reflection v+ then is

v+ = v−⊥ − v−‖

= v− − 2(v− ·n0

)n0 . (3.7)

To determine the next reflection point for a given initial condition r0,v0, we parametrisethe line along which the particle moves by

r(t) = r0 + tv0 . (3.8)

Intersecting r(t) with the scattering obstacles given by (3.4), we now have

(r(t)− ri)2 = R2 . (3.9)

Inserting the expression for r(t) and solving (3.9) for t yields

t± =−v0 · (r0 − ri)±

√((r0 − ri) ·v0)2 − (r0 − ri)2 +R2

v20

. (3.10)

For each sphere denoted by the index i, this equation of second degree has either 0, 1 or2 solutions, depending on the discriminant

D = ((r0 − ri) ·v0)2 − (r0 − ri)2 +R2 . (3.11)

Whereas a negative discriminant means that the i-th sphere is not hit by the particle, avanishing discriminant means that the sphere is touched by the particle’s trajectory1,and positive D means that the line r(t) intersects the i-th sphere twice. The two pointsobtained by either choosing t+ or t− describe a reflection on the sphere’s “forefront” or onthe “rear side”. As the particles must not cross the scatterers, we can rule out t+ becauseit leads to a greater distance between the reflection points. This case describes reflection

1This case is practically ruled out due to finite numerical precision.

11


at the “rear side”. The case of negative t is also mathematically possibly. This case hasto be ruled out because a scattered particle must not travel “backwards”. After checkingall i = 1, . . . , N scattering obstacles, the closest reflection point is obtained. This pointas well as the new velocity v+ is then used to check for the next reflection. If no sphereis hit, we know that the particle has escaped towards infinity. Iterating this process aslong as the particle stays in the scattering system, a sequence of reflection points thatuniquely describes a particle’s trajectory is obtained. Track may also be kept of the orderin which the single spheres are hit; a symbolic code can be assigned to the particle’s “fate”determined by the initial condition. If each of the spheres is labelled by a character fromthe alphabet ωi = A,B,C,D, a trajectory that experiences N reflections corresponds toa sequence ω1 . . . ωN . These sequences will be used in the estimation of the repeller.

3.3 Fractal repeller

The repeller is the set of trajectories that stays trapped for t → ±∞. As shown insection 2.2, it is the union of the global stable manifold Ws and the global unstablemanifold Wu. As time reversibility implies that both have the same fractal dimension,only the stable manifold Ws needs to be considered, i.e. we are searching for trajectoriesthat stay trapped for t→ +∞. Below, the time-delay function T will be introduced asa convenient measure to work with. The structure of T will be worked out for thosebilliards that have been studied in this thesis.

3.3.1 Time-delay functions

As the time a particle spends in the billiard before escaping towards infinity dependson the magnitude of the velocity v, it is more convenient to use the time-delay function,i.e. the number of reflections T as a measure of time spent in the scattering system.The stable manifold Ws of the repeller is given by the phase space points for whichthe time-delay function T is singular. The reflection count T does not depend on themagnitude of v. It maps the continuous set of initial conditions onto an integer number.The discrete nature of T will prove useful for the computation of the Hausdorff dimensiondH, as will be discussed in section 3.6.

Time-delay functions for the three-disk billiard

As the three-disk billiard has only 2 spatial dimensions, it will be discussed briefly forillustration. In the three disk billiard, we may choose the initial conditions along aone-dimensional line, e.g. x = 0. This represents a PSOS with sectioning condition x = 0.As the magnitude of the momenta p is constant because of conservation of energy, it maybe chosen arbitrarily. Conditions for the momenta, e.g. px = 1 and py = 0 form othersurfaces of section. Sample trajectories with this choice of initial conditions are drawn

12

3.3 Fractal repeller

0

2

4

6

8

10

12

14

0 0.2 0.4 0.6 0.8 1

T

λ

(a)

0

2

4

6

8

10

12

14

0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3

T

λ

(b)

Figure 3.2: Time-delay function T against a parameter λ. The parameter specifies thestarting point r = r0 + λr of the trajectory. A detailed illustration of the interval[0.2; 0.3] of (a) is shown in (b). The self-similarity of the fractal repeller can be seenin these figures.

in figure 2.1. A sample time-delay function and a magnification thereof are shown infigure 3.2.

The structure of the time-delay functions can be understood from the geometry of thebilliard: there is a parameter interval that corresponds to initial conditions that hit thefirst disk. Within this interval, there are two more non-connected intervals that leadto T ≥ 2 reflections – each interval belonging to encounters with one of the two otherdisks. The same statement holds for any number of reflections. Although the repellercontains the periodic orbits only as a subset, the structure in time-delay functions hasthe same origin as the existence of symbolic dynamics for the periodic cycles, for furtherdiscussion see section 4.2.

Time-delay functions for the four-sphere billiard

Whereas in the three-disk billiard only one parameter was sufficient to measure time-delayfunctions, there is only enough fractal structure contained in the time-delay functionsof the four-sphere billiard for large centre-to-centre separations if a two-dimensionalPSOS is chosen. For the calculations performed in this thesis, the velocities for theinitial conditions are chosen parallel to the z-axis, v = (0, 0, 1). The spatial positionsto iterate from are chosen in a plane perpendicular to the z-axis, z = z0. For thebilliard configuration given by equations 3.1, we choose z0 < 0 such that particles enterthe tetrahedron from “below“, i.e. through the plane spanned by three spheres whosecentres are at zi = 0. The resulting time-delay functions are maps of the set x0, y0 tonon-negative integers N0. A sample time-delay function as well as a blow-up indicatingthe fractal structure is shown in figure 3.3.

13


-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

x

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

y

2

3

4

5

(a)

-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02

x

0.105

0.11

0.115

0.12

0.125

0.13

0.135

0.14

0.145

y

2

3

4

5

6

7

(b)

Figure 3.3: (a) Time-delay functions in the range 2 ≤ T ≤ 5 for d/R = 3 and (b) amagnification thereof in the range 2 ≤ T ≤ 7 . The time-delay functions are drawnas functions of the coordinates x, y in the surface of section. The colours indicate thevalue of T . Only the boundaries of individual regions are drawn; initial conditionschosen within the regions experience the same number of reflections. Apparently,the boundaries of the individual regions may be approximated by ellipses. This factwill prove useful for the approximation of the repeller. Analogous to the three-diskbilliard, the self-similarity suggests that the stable manifold is a fractal set.

Analogous to the three-disk billiard the time-delay functions exhibit a certain structure.Within each region of a given reflection count, there are three other regions of higherreflection count. Each of these regions corresponds to consecutive encounters with one ofthe three other spheres. If the ratio d/R is large enough, there is a circle in the Poincarésurface of section that encloses all initial conditions that visit the uppermost sphere first.This circle is the projection of the sphere onto the surface of section. Within this circle,there are three more regions of initial conditions that experience a higher number ofreflections. From figures 3.3 it is evident that for small d/R the boundaries of each regionmay well be approximated by ellipses.

Besides the two-dimensional calculations, one might choose random one-dimensionalcuts in the surface of section. However, these cuts only contain enough information onthe fractal repeller if a “good” cut is chosen [17]. This can be understood by the factthat for larger separations in the four-sphere-billiard, the dimension D of the repellercalculated from the stable manifold Ws in a two-dimensional PSOS might drop below1. In this case, an additional cut that lowered the dimension by 1 once more would notyield fractal structures a sensible calculation of dH can be performed for. A colourful

14

3.4 Hausdorff dimension

Figure 3.4: Picture obtained from ray-tracing in a system of four touching sphereswith reflective surfaces and sources of coloured light placed outside the tetrahedron.The self-similar structures indicate the fractal property of the repeller. An actualphotography similar to this computer simulation figure can be found in [6, 18].

illustration of the fractal resulting from chaotic scattering in the four-sphere billiard withtouching spheres is shown in figure 3.4.


As was illustrated in the preceding section, with increasing reflection count T the intervalsrespectively regions decrease in size. As the stable manifoldWs is the union of all regionswith infinite T , it is evident that the repeller does not fill the whole surface of section.The stable manifold resembles a “sponge” with “holes” of finite T . Such objects thatadditionally exhibit the property of self-similarity are called fractals.

The familiar concept of dimension is not suitable to characterise fractals. However, itis possible to define a new notion of dimension: The so called Hausdorff dimension orfractal dimension generalises the concept of dimension to “perforated” sets such as thestable manifold in billiards. Consider a set S embedded in Rd. Cover the set S by N(ε)d-spheres of radius ε. The number of balls needed for minimum coverage scales like

N(ε) ∝ 1

εdH. (3.12)

15


11/3

1/9 ...

Figure 3.5: The middle third Cantor set is constructed by step-wise removal of themiddle third of each interval starting from the interval [0; 1]. The Cantor set is theset of points left after iterating ad infinitum. The Hausdorff dimension calculatedfrom the definition (3.13) is dH = lim

n→∞log(2n)/ log ((1/3)n) = log(2)/ log(3) .

Solving this equation for dH in the limit of ε→ 0 yields the definition of the Hausdorffdimension

dH = dim(S) := − limε→0

logN(ε)

log ε, (3.13)

which may be a non-integer number. For sets that fill the embedding set Rd densely, itcan be seen from the definition that the usual spatial integer dimension d agrees withthe fractal dimension dH.It is possible to compute dH directly from the definition (3.13) using so called box-

counting algorithms. Here, the set to measure is covered with d-balls of decreasing size εwhilst the minimal number N(ε) of balls needed to cover the set is measured. For theCantor set this procedure is illustrated in figure 3.5. The box-counting method, however,is not suited for the billiards under consideration as it requires iteration of a vast amountof initial conditions on a grid. The regions of high T that exhibit fractal properties maynot be resolved with acceptable computational effort. A method better suited to billiardswill be introduced below.

3.4.1 Estimating dH through Hausdorff sums

The time-delay function T may be used to approximate the Hausdorff dimension of therepeller. We only use the stable manifold Ws. First, we introduce the auxiliary quantityl(i)n that denotes the length of the n-th interval of initial conditions with T ≥ i reflectionsin the surface of section. For line-like time-delay functions as in the three-disk billiard, theinterval lengths are just lengths of line-segments; whereas in three-dimensional billiards,a two-dimensional PSOS may be chosen. The quantities l(i)n are then areas of regionscontaining initial conditions leading to T ≥ i bounces. Define the quantities

K(i)(s) :=∑n

(l(i)n)s

(3.14)

16


0

0.5

1

1.5

2

2.5

3

3.5

0.5 0.55 0.6 0.65 0.7 0.75

K(s

)

s

Hausdorff sums for the Cantor set

log(2)/log(3)

i=2i=4i=6i=8

Figure 3.6: Hausdorff dimension for the Cantor set, determined from intersected Haus-dorff sums in comparison to the exact value of dH = log(2)/ log(3) indicated by thearrow. After iteration i, the interval length l(i) is given by 2i(1/3)i. As the intervallengths are analytically known, the approximation of dH is exact in this case even forlow i. In real billiards, the fractal structure of the time-delay functions T may notbe pronounced in intervals of small i.

which will be called Hausdorff sums below. These sums have the following properties[19, 20]:

limi→∞

K(i)(s) =

∞ for 0 ≤ s < dH

const. > 0 for s = dH

0 for dH < s <∞(3.15)

The property for s = dH stems from the fact that the Hausdorff sums by definitionare smooth functions of the variable s. This allows us to use K(i)(s) to estimate theHausdorff dimension dH. Finite numerical precision and finite computing time availableprevent the determination of initial conditions that lead to trapped orbits. Instead, itis possible to estimate the Hausdorff dimension from intervals of finite T . IntersectingK(i)(s) for different i approximates the Hausdorff dimension [17, 7]. An illustration ofthis method for the Cantor set is shown in figure 3.6.The values of the Hausdorff sums themselves do not provide any information. The

only useful quantity that may be extracted from plots such that in figure 3.6 is the

17


intersection point of curves for different T . Furthermore, from the definition given above,it is evident that scaling r → γr implies

K(i)(s)→ γsK(i)(s) , (3.16)

which does not alter the intersection point of Hausdorff sums for different numbers ofreflection i. This once more justifies that only the ratio d/R is of interest, whereas theindividual choice of each d and R is irrelevant.

3.5 Algorithm

Existing methods of estimating the repeller’s fractal dimension have been limited to anarrow range of the parameter d/R, specifically to the case of almost-touching spheres[5, 6, 17, 7]. The algorithm discussed in the following paragraphs allows calculationsfor a wider range of d/R. To estimate the stable manifold Ws in the PSOS, it will benecessary to accurately find boundaries of regions of a given symbolic code ωi . . . ωN andreflection count T = N .

3.5.1 Interval bisection in billiards

As preliminary work for the algorithm, an interval bisection algorithm suited to thesituation in billiards will be discussed briefly. By interval bisection, boundaries to regionsof specific T can be found.

Presume a point r1 on the surface of section with T = T1 and symbolic code ω1 . . . ωN1

and another point r2 with T = T2 and symbolic code ω1 . . . ωN2 both are known. Fur-thermore, presume T2 > T1. On the line connecting both points there has to be anotherpoint r at which the time-delay function T discontinuously changes its value from T1 toT2. Bisection of a function is based on the evaluation of the function in the middle of theinterval, in this case iterating a trajectory from

r =r1 + r2

2. (3.17)

The symbolic code obtained from this iteration is ω1 . . . ωN . If

T (r) = T1 , (3.18)

the middle point lies within the region that also contains r1. In this case, the next stepin the bisection is to replace r1 with r. However, the case

T (r) = T2 (3.19)

has to been handled carefully; compare figure 3.7. If the number of reflections at r2 andr coincides, another region of T2 enclosed in the one of T2 may have been found. If so,

18

3.5 Algorithm

Figure 3.7: Illustration of the bisection problem. The region drawn in pale red containsinitial conditions that lead to T1 reflections; iterating from regions drawn in darkerred, however, yields T2 > T1 reflections. From the sketch it is evident that the pointin the middle of the dark crosses labelled with “Pos. 1” respectively “Pos. 2” mayexperience the same number of reflections as the point at the label “Pos. 2”. Forthis reason, it is crucial to use the symbolic code to distinguish the regions drawn indarker red. As the symbolic code corresponding to initial conditions enclosed in thoseregions differs at the T2-th position, each of the dark red regions can be identifieduniquely.

the T2-th character in the symbolic codes for r2 and r differs. In this case the next stepis also to replace r1 by r. In any other case, r is already contained in the same regionas r2, and r2 has to be set to r. The whole procedure is repeated until the distancebetween points r1 and r2 has dropped below a given small value ε,

|r1 − r2| < ε . (3.20)

Then, finally, the jump discontinuity of T , viz. the boundary point of the region containingr2, has been determined with an absolute error of ε. With this kind of bisection, itis possible to determine all jump discontinuities of T . In each step of the bisection,the absolute error ε shrinks by a factor of 2. Therefore, the estimate improves byapproximately one order of magnitude each 3 steps of iteration.

3.5.2 Finding regions

Assumptions

The algorithm used in this thesis relies on the structure of the time-delay functionsdiscussed in subsection 3.3.1. In the calculations, the following assumptions are made:

• Regions of a specific order of visits with the scatterers described by the symboliccode are non-overlapping.

19


-0.15 -0.1 -0.05 0 0.05 0.1 0.15

x

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

y

2

3

4

5

(a)

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

x

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

y

2

3

4

(b)

Figure 3.8: Boundaries to regions of different reflection counts T for (a) d/R = 4 and(b) d/R = 2.5. Polygonal boundary chains are drawn with crosses, whereas solid linesrepresent fitted ellipses. In (a), there is perfect agreement of ellipses and polygonalchains. However, in (b), the boundaries of regions with low T are clearly of suchshape that an approximation by ellipses is not suitable anymore. It also can be seenfrom (b) that the distance of individual regions shrinks. If d/R is increased evenmore, the regions start to touch and finally, overlap. In this case, our assumptionsdo not hold anymore and the procedure presented here fails.

• Within a region of a given order of visits, there are exactly three more regions eachcorresponding to additional visits at one of the three other spheres.

Both conditions may be violated for d/R close to 2, i.e. the case of almost touchingspheres. The tendency to violate the conditions for d/R approaching 2 can be seen fromfigure 3.8.

Procedure

All steps are based on the Poincaré surface of section z = z0 chosen such that iterationstarts from a plane parallel to the plane spanned by the three closest spheres. Thevelocities are chosen parallel to the z-axis such that the uppermost sphere is visited first.

Assume that regions with Tmin ≤ T ≤ Tmax are sufficient for an estimation of dH.Under this assumption, it is possible to find regions approximating the repeller with thefollowing procedure.First, the projection of the uppermost sphere in the billiard – see the coordinate

tuple indexed by 1 in equations (3.1) – onto the surface of section is determined. Forsmall d/R, this region is a circle, for larger d/R, projections of the other spheres may

20

3.5 Algorithm

be cut out of the circle. This is done by randomly choosing a point in the surface ofsection and a bisection between this point and points equally distributed on a large circlefully containing the projection of sphere 1. By assumption, inside this region there arethree other regions with T = 2. The corresponding symbolic codes differ in the secondcharacter.One possibility to store the boundaries is by holding a polygonal chain. As in this

procedure, the number Nregions of regions grows exponentially with

Nregions = 3T−1 , (3.21)

this way of data storage is memory-expensive. An alternative is to approximate theboundary by an ellipse.An ellipse is described by the polynomial

a1x2 + a2y

2 + a3xy + a4x+ a5y = 1 , (3.22)

which can be fitted to the polygonal boundary. The fit parameters a1 . . . a5 containinformation on the semi-major axes, the rotation and the centre shift of the ellipse.Details on the extraction of the ellipse’s parameters will be given below. For largernumber of reflections, the ellipses may be fairly small. In this case, the shift of the ellipse’scentre from the origin of the coordinate system can be several orders of magnitude largerthan the semi-major axes. In this case, least-square fitting can fail as the terms containinginformation on the shift dominate terms containing information on the semi-major axes.To prevent this, the polygonal chain can be shifted by r0. After successful fitting, theellipse is shifted back again. The fitted ellipses allow analytical calculation of the area.After the boundary of the region with T = 1 has been determined, randomly chosen

initial conditions within this region are iterated to find three regions of T = 2. Oncethese are found, polygonal chains forming boundaries to each of the new distinct regionsare calculated. Again, ellipses are fitted. This procedure is iterated until all desiredregions corresponding to Tmin ≤ T ≤ Tmax have been found.

To improve accuracy, it is now possible to shift the region’s midpoints to the midpointof the ellipses. All bisections for the polygonal chain boundaries are repeated in such away that all lines connecting the ellipse’s midpoints and the boundary points intersect atidentical angles. This will be beneficial in the construction of Hausdorff sums.

3.5.3 Building Hausdorff sums

Once all desired boundaries have been calculated, the fractal dimension dH can beestimated. To build the Hausdorff sums (3.14), the areas enclosed in the individualregions have to be known.

21


Areas from ellipses

From the ellipses fitted to the boundaries, the area A is trivially given by

A = πab , (3.23)

where a and b are the semi-major axes of the ellipses. The parameters a, b, the centreshift (x0, y0) as well as the rotation angle ϕ can be extracted from the polynomial (3.22).First, the terms originating from the shift (x0, y0) have to be eliminated from (3.22).Considering the polynomial as a quadratic equation in x, the largest and smallest valuesof y, y1,2, can be obtained by setting the discriminant Dx

Dx =(a3y

2 + a4

)2 − 4a1

(a2y

2 + a5y − 1)

= 0 . (3.24)

For the case of Dx = 0, the curve described by (3.22), f(x = 0, y) = 0 is a parabola withthe two roots y1 and y2. These are the largest respectively the smallest values of y. Theellipse’s centre coordinate y0 can be obtained from the equation

y0 =y1 + y2

2, (3.25)

which evaluates to

y0 =2a1a5 − a3a4

a32 − 4a1a2

. (3.26)

Similarly, from

Dy =(a3x

2 + a5

)2 − 4a2

(a1x

2 + a4y − 1)

= 0 , (3.27)

the second centre coordinate x0 is found to be

x0 =x1 + x2

2=

2a2a4 − a3a5

a32 − 4a1a2

. (3.28)

A shift of the coordinate origin to (x0, y0) yields the second degree equations

A1x2 + A2y

2 + 2A3xy = 1 , (3.29)

which contains no linear terms in x and y. Equation (3.29) can be written as the quadratic

[x y]

[A1 A3

A3 A2

] [xy

]= 1 . (3.30)

The 2× 2 matrix in this equation may be diagonalised by a rotation,[A1 A3

A3 A2

]= O

[1/a2 0

0 1/b2

]Ot (3.31)

22

3.5 Algorithm

with O the orthogonal rotation matrix

O =

[cosϕ − sinϕsinϕ cosϕ

]. (3.32)

From these considerations, explicit expressions for the ellipse’s parameters are given by

x0 =2a2a4 − a3a5

a32 − 4a1a2

, (3.33a)

y0 =2a1a5 − a3a4

a32 − 4a1a2

, (3.33b)

tanϕ = − a3

a1 − a2 +√

(a1 − a2)2 + a32, (3.33c)

a2 =2 (a2a4

2 − a3 (a3 + a4a5) + a1 (4a2 + a52))

(4a1a2 − a32)(a1 + a2 +

√(a1 − a2)2 + a3

2) , (3.33d)

b2 = − 2 (a2a42 − a3 (a3 + a4a5) + a1 (4a2 + a5

2))

(4a1a2 − a32)(−a1 − a2 +

√(a1 − a2)2 + a3

2) . (3.33e)

Once the semi-major axes a and b are known, the area of the ellipse, again, equalsA = πab.

Areas from polygonal chains

The areas enclosed by the polygons are easily calculated using numerical quadrature ofthe area given by

A =1

2

2π∫0

r2(ϕ) dϕ (3.34)

using the midpoint or the Simpson rule. The term r(ϕ) denotes the distance of theboundary point from the “midpoint” of the region. An uneven number of supportingpoints yields better results, which can be understood from the fact that for ellipse-likeshapes, there is an approximate rotational symmetry about an angle of π. For an evennumber of supporting points, r(ϕ) and r(ϕ + π) are approximately equal and detailsabout the shape of the boundary are lost. A fairly low number of supporting points hasproved to be sufficient for very high precision. All calculations have been performed with101 supporting points.

23


d/R 2.5 2.75 3 3.5 4 4.5

dH 0.4774 0.4205 0.3818 0.3314 0.2992 0.2766

d/R 5 6 7 8 9 10

dH 0.2596 0.2354 0.2184 0.2063 0.1966 0.1888

Table 3.1: Numerical values of the Hausdorff dimensions dH of the stable manifold Ws

for various configuration parameters d/R. All decimal digits are significant.

3.6 Results

Calculations for d/R = 2.5 to d/R = 10 have been performed. Sample plots for intersectedHausdorff sums K(i)(s) are shown in figures 3.9. Results for the Hausdorff dimension dH

are compiled in figure 3.10 and in the table 3.1. The calculations using polygonal chainsagree up to four decimal digits with the calculations using fitted ellipses.Figure 3.10 clearly shows that with decreasing d/R the intersection of the stable

manifold Ws with the Poincaré surface of section fills the plane denser. The repeller’sdimension dH thus increases as the tetrahedron gets packed more densely.

In conclusion, the method presented here establishes a fast and very precise method ofgauging the repeller. Though the assumptions are quite strong, they hold over a widerange of the ratio d/R.

24

3.6 Results

-0.85

-0.8

-0.75

-0.7

-0.65

-0.6

-0.55

-0.5

-0.45

-0.4

-0.35

0.23 0.231 0.232 0.233 0.234 0.235 0.236 0.237 0.238 0.239 0.24

Lo

garith

m o

f H

ausd

orf

f su

ms

s

T=4T=5T=6T=7T=8T=9

T=10

(a)

-0.85

-0.8

-0.75

-0.7

-0.65

-0.6

-0.55

-0.5

-0.45

-0.4

-0.35

0.23 0.231 0.232 0.233 0.234 0.235 0.236 0.237 0.238 0.239 0.24

Logarith

m o

f H

ausd

orf

f su

ms

s

T=4T=5T=6T=7T=8T=9

T=10

(b)

Figure 3.9: Intersected Hausdorff sums K(i)(s) for various reflection numbers i = Tcalculated from (a) a polygonal chain with 101 supporting points and (b) ellipsesfitted to polygonal chains for d/R = 6. As can clearly be seen, the intersection pointsfor all shown curves agree perfectly. For this reason, the Hausdorff dimensions canbe given to at least 4 decimal places, which corresponds to a relative precision ofabout 1 · 10−4.

25


0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

2 3 4 5 6 7 8 9 10

Hausdorf

f dim

ensio

n d

H

d/R

Figure 3.10: Hausdorff dimension dH of the stable manifoldWs for various ratios of d/Rin the interval [2.5; 10]. All data points have been obtained by intersecting Hausdorffsums as demonstrated in figures 3.9. Using a hyperbola as a fitting function, weextrapolate d/R = 2.461 as the critical value at which the stable manifoldWs reachesthe dimension dH = 0.5 . From the critical point on, the dimension of the repeller ishigher than 1 and one-dimensional cuts could be used for calculations.

26

4 Semiclassical quantisation ofquantum billiards

Studying a billiard system in a purely quantum mechanical fashion turns out to becomplicated as – although particles move freely in between the scatterers – it is a demand-ing task to find wavefunctions that vanish on all scatterer’s boundaries simultaneously.For attempts on N -sphere scattering systems in three dimensions, see [21]; for the two-dimensional three-disk scattering system see [22]. Results and comparisons of methodsfor the four-sphere scattering system are presented in [8]. The techniques of semiclassicalquantisation presented below are better suited for billiard systems.

4.1 From the trace formula to zeta functions

4.1.1 The trace formula

As one of the great achievements of semiclassical physics, Gutzwiller’s trace formulaprovides a mean of quantising a system via periodic orbits [23]. The density of statesg(E) may be expressed by the trace of the Green’s function G(E + iε). Gutzwiller’sderivation is based on the relation

g(E) =∑n

δ(E − En)

= − 1

πIm trG(E + iε)

= − 1

πIm

∫d3r G(q, q;E + iε) , (4.1)

which connects the density of states g(E) to the trace of the system’s Green’s functionG(E). For bound systems, Green’s function, defined by

G(E) =∑n

1

E − En, (4.2)

27

4 Semiclassical quantisation of quantum billiards

has poles along the real energy axis. Each pole corresponds to an energy eigenvalue.Expressing the Green’s function as the Fourier transform of the propagator K(q′, q′′, t)and using the Feynman path integral propagator [24]

K(q′, q′′, t) =

∫Dq exp

i

~

t∫0

L(q, q, t′) dt′

, (4.3)

Gutzwiller showed that formally g(E) may be expressed in terms of classical periodicorbits and their properties. For details on the derivation, compare e.g. [23, 25, 26]. Thederivation is based on repeated applications of stationary phase approximations for allintegrals with rapidly oscillating integrands. The first approximation requires stationaryaction. Thus, only classical paths contribute to the density of states g(E). In the nextstep, as the calculation of the trace requires q′ → q, only closed orbits contribute tothe density of states. Further stationary phase approximations narrow the number ofcontributions to g(E) even further down – it is found that only periodic orbits enterinto the result. The distinction between orbits of zero length and orbits of finite lengthrequires to split the density of states into two parts,

g(E) = g(E) + δg(E) . (4.4)

The term g(E) is smoothly varying with energy, whereas δg(E) is fluctuating with theenergy E. Details on the spectrum are contained in the fluctuating part. Only finitelength periodic orbits contribute to the fluctuating part, the zero-length orbits contributeto the smooth part. The integrated smooth part is a Weyl law in the sense of section 2.1.A derivation of the integrated density of states

N(E) =

E∫0

g(E ′) dE ′ (4.5)

for billiard systems is given in that section. For this thesis, the counting function N(k)is calculated from the detailed spectra.

In the semiclassical limit, where all actions are large compared to ~, the semiclassicalapproximation is applicable. Gutzwiller found

δgscl(E) =1

~π∑p

∞∑r=1

Tp√|det(Mp,r − 1)|

exp

[i

(1

~Sp,r − σp,r

π

2

)]. (4.6)

to be the expression for the fluctuating part of the density of states if all periodic orbitsare isolated, i.e. not connected to each other by continuous transformations. By the traceformula the density of states g(E) is expressed as a Fourier decomposition depending onlyon properties of classical primitive periodic orbits p and their r-times repeated revolutions.

28


In (4.6), Sp,r denotes the action of a periodic orbit, whereas Tp is the primitive periodicorbit’s period. The reduced monodromy matrix Mp,r provides information on the linearevolution of a small aberration from an initial condition belonging to a periodic orbitover one period [25]. It may be calculated in a Poincaré surface of section perpendicularto the orbit. The eigenvalues of Mp,r quantify the stability of the periodic orbit. Due tothe symplectic structure of Hamiltonian mechanics, the eigenvalues come in tuples

λ,1

λ, λ∗,

1

λ∗. (4.7)

For further details, see [25]. The quantity σp,r stands for the so called Maslov indexbelonging to a certain orbit. The Maslov index may be understood as a book-keepingvariable keeping track of the phase shifts the wavefunction experiences by reflectionsat potential barriers. Further mathematical details on Maslov indices in a geometricalcontext may be found in [27]; for a general discussion of Maslov indices in the traceformula, see [28]. For billiard systems with Dirichlet boundary conditions, compare thediscussion in [26].

Although the trace formula is a powerful tool in semiclassical physics, it is plagued byserious convergence problems. In chaotic systems, the number of periodic orbits typicallygrows exponentially with length l. This growth usually cannot be compensated by thedecrease of the amplitude factors [29]

Ap,r =Tp√

|det(Mp,r − 1)|. (4.8)

One method of improving the convergence ideally suited for billiard systems is cycleexpansion of zeta functions. This method will be introduced in the following subsections.

Note that in parts of the literature, only bound systems are discussed. In such systemsall energies are real and the oscillating part of g(E) could have been written with acosine instead of the complex exponential function. However, the trace formula may beanalytically continued to the complex plane. It then – formally at least – also holds foropen system with complex resonances. In the following we will drop the restriction toclosed systems.

4.1.2 The trace formula for billiards

For billiard systems, it is convenient to use scaling properties of the trace formula torewrite it as an equation depending on the wavenumber k. We will inspect the singleterms in the trace formula (4.6) to come to a reformulation.

1. First, it has to be noted that the calculation of Maslov indices for systems withsmooth potentials may be tedious. However, we may simplify the phase term in

29


the trace formula. For repeated revolutions of unstable orbits the Maslov indicesare additive,

σp,r = rσp . (4.9)

In billiard systems with Dirichlet boundary conditions, the wavefunction changessign respectively collects an extra phase shift of π with every reflection. This ismirrored in the Maslov indices as well. Equation (4.9) may thus be written

σp,r = 2nr , (4.10)

where nr is the number of reflections. Using the additive property, the Maslovindex reads

σp,r = 2rnp . (4.11)

2. Just as the Maslov indices, the actions for repeated revolutions

Sp,r = rSp . (4.12)

are additive as well.

3. With the velocity v and the physical cycle length lp, the period Tp can be expressedas

Tp =lpv

=mlp~k

. (4.13)

4. The stability properties of r-times repeated revolutions can be obtained from thecorresponding primitive cycle,

Mp,r = (Mp)r . (4.14)

5. Finally, as the number of states is independent of the choice of variable,

g(k) dk = g(E) dE , (4.15)

the energy E may be eliminated from the density of states in favour of the wavenum-ber k,

g(k) =dE

dkg(E) =

~2k

mg(E) . (4.16)

30


Collecting all results of (4.9) - (4.16) and inserting them into the trace formula (4.6)then gives the trace formula for billiards,

δgscl(k) =1

π

∑p

∞∑r=1

(−1)rnp lp√|det(Mp

r − 1)|exp [irlpk] . (4.17)

An alternative form which will be used in the following is obtained by explicitlyusing the dimensionality of the problem. In three spatial dimension, the phase space issix-dimensional. The full monodromy matrix thus has six eigenvalues [25]. Two of themare 1. One of those eigenvalues corresponds to initial aberrations along the periodic orbit,the other one corresponds to the choice of a different energy, which is stable due to theconservation of energy. The reduced monodromy matrix for the r-th revolution of theprimitive cycle p, Mr, lacks those eigenvalues and det (Mp

r − 1) thus can be written asthe product

det (Mpr − 1) = (1− Λr

1)(1− Λ−r1

)(1− Λr

2)(1− Λ−r2

)=(2− Λr

1 − Λ−r1

) (2− Λr

2 − Λ−r2

)(4.18)

where Λ1 corresponds to λ in the tuple of eigenvalues given in (4.7), and Λ2 correspondsto λ∗. With these substitutions, the trace formula (4.17) reads

δgscl(k) =1

π

∑p

∞∑r=1

(−1)rnp lp e(irlpk)√∣∣(2− Λr1 − Λ−r1

) (2− Λr

2 − Λ−r2

)∣∣ . (4.19)

The trace formula allows an interpretation of quantum spectra in terms of periodicorbits [30]. By Fourier transformation, we have

δg(l) =

∫δg(k) eikldk

=∑p

∑r

Ap,rδ(l − rlp) . (4.20)

Each cycle of length rlp leads to a peak in δg(l). These peaks are weighted with theamplitude factor (4.8).

4.1.3 The Gutzwiller-Voros zeta function

Voros proposed another formulation of the density of states [31] by introducing a function

Z(k) =∏n

(k − kn) (4.21)

31


whose zeros correspond to the quantum eigenvalues kn. The relation with the densityg(k) arises by taking the logarithmic derivative

g(k) = − 1

πIm

d

dklnZ(k)

= − 1

πIm∑n

d

dkln(k − kn)

= − 1

πIm∑n

1

k − kn + iε, (4.22)

which is just the density of states g(k) in (4.1). For billiards, the Gutzwiller-Voros zetafunction equals

ZGV(k) = exp

(−∑p

∞∑r=1

1

r

(−1)rnp eirlpk√|det(Mp

r − 1)|

). (4.23)

Equivalency to (4.17) can be shown by calculating the logarithmic derivative as demon-strated above.

4.2 Symbolic dynamics and periodic orbits in billiards

As demonstrated above, semiclassical quantisation relies on periodic orbits. The methodof cycle expansion requires all periodic orbits up to a given maximum cycle length nr. Inspecial billiards it is convenient to assign a so called symbolic code to each periodic orbit.These codes will be introduced in this section and the numerical periodic orbit searchwill be discussed briefly.

4.2.1 Full symbolic codes

Symbolic codes assign a set of characters from a given alphabet ω1, . . . , ωn to atrajectory. In billiards, a label ωi for each scatterer may be chosen. For example, in thefour-sphere scattering systems, we may label the individual spheres with

ωi = A,B,C,D . (4.24)

A sequence of symbols from the alphabet then provides information on the visitationsequence, i.e. the order in which the scatterers have been hit by the particle before it leftto infinity. Because the same scatterer cannot be visited twice in row, the only restrictionon the sequence is that a specific label must not appear in two or more consecutive placesin the symbolic code. Also note that in a billiard with N scatterers, N characters areneeded in the alphabet.

32

4.2 Symbolic dynamics and periodic orbits in billiards

Symbolic codes for periodic orbits

Periodic orbits are conveniently described by a symbolic code. As an example, thesymbolic code for the periodic orbit visiting spheres A and B in turns is

. . . ABABABABAB . . . . (4.25)

This code consists of periodic repetitions of the smallest building block AB. For thisreason, the notation can be shortened to

AB . (4.26)

A periodic orbit is then uniquely given by a bi-infinite sequence of characters from thechosen alphabet. In the following, the bar indicating periodicity will be dropped. Formost cases, any sequence of those characters corresponds to one periodic orbit. If thisone-to-one correspondence is not given, i.e. if some orbits become unphysical because theypenetrate the scatterers, we speak of pruning. This is the case for small centre-to-centreseparations. In the four-sphere scattering system, the symbolic dynamics has beenshown to be pruned for configuration parameters below d/R = 2.0482 [8]. The lengthrespectively the number of characters in a symbolic code in the sense of equation (4.26)is called the cycle length n. If there is no pruning, the number N of scatterers determinesthe number of periodic orbits of a given cycle length l . After each collision, there areN − 1 different scatterers allowed in the next collision. Thus, the number of allowedcycles grows exponentially with the cycle length. The topological entropy htop is

htop = ln(N − 1) (4.27)

per symbol. Instead of a formulation “per symbol”, we may also express htop for theorbit’s lengths. The number of orbits proliferates exponentially with ehtopl as l increases.For convergence of the periodic orbit sum (4.6), this proliferation has to be compensatedby the exponentially decreasing amplitude factors. The topological entropy leads to theso called entropy barrier Im k = h, above which the trace formula is well-defined [29].

4.2.2 Symmetry reduction

In systems with discrete symmetries such as the four-sphere billiard, which is invariantunder all symmetry operations of the tetrahedron group Td, whole classes of symboliccodes are equivalent to each other. For example, in the four sphere billiard, the 6 orbitslabelled by AB,AC,AD,BC,BD,CD can be mapped onto each other by the symmetryoperations of Td. Furthermore, cyclic permutation of the characters ωi leaves the orbitsinvariant. For these reasons, it is appropriate to use the symmetry properties to introducethe following short notation. First, define the plane of reflection as the plane that containsthe centres of the last three distinct spheres visited. Then, instead of labelling all spheresindividually, the label 0 will be used if the orbit visits the last sphere once more, 1 will

33


p p hp

0 AB σd, C2

1 ABC C3

2 ABDC S4

01 ABAC σd02 ABADAC C3

12 ABCDBADC S4

Table 4.1: Symbolic code p of the shortest symmetry reduced periodic orbits and theircorrespondence p in full physical space if the spheres A,B are chosen to be the firstin the visitation sequence. Also, the symmetry class hp of the orbits is given (from[8]). Details on the symmetry of the system are given in subsection 4.4.2.

indicate a visit at the third other sphere in the same plane of reflection, whereas thelabel 2 will be used for a visit at the fourth sphere outside the plane of reflection. Withthis nomenclature, all symbolic codes containing the character 2 are three-dimensional,whereas orbits corresponding to sequences of 0 and 1 are two-dimensional. The newlabelling reduces the number of characters in the alphabet to three, i.e. the code isternary. This reduction corresponds to a reduction of the full physical phase space Mto the fundamental domain M from which the whole phase space can be reconstructedby applying the symmetry group’s elements. If one chooses the plane spanned by thespheres A,B,C as the reflection plane and AB as sequence of the first two visits, thenthe reduced symbolic codes correspond to physical orbits as shown in table 4.1. Notethat the symmetry reduced orbits are in general shorter than the corresponding physicalones. Only the symmetry reduced orbits that have the identity operation E as maximumsymmetry have the same length as the corresponding physical orbits. The reduced orbitsof symmetry classes σd and C2 yield twice as long physical orbits, C3 orbits are threetimes as long in the physical space, and, finally, S4 orbits have quadruple length.

4.2.3 Finding periodic orbits

Semiclassical quantisation generally relies on the knowledge of periodic orbits. For cycleexpansion in particular, all periodic orbits have to be known up to a given cycle length.Furthermore, the symbolic dynamics has to be complete, i.e. every symbolic code has tocorrespond to one periodic orbit. In the case of pruning, this requirement is not met.If complete symbolic dynamics is the case, the search for periodic orbits can be

accomplished by using Hamilton’s principle

δS = 0 . (4.28)

The symbolic code determines which scatterers are visited by the trajectory as well asthe order of the visits. Moreover, for free particles in billiards, the action S belonging to

34

4.3 Cycle expansion of zeta functions

a given trajectory is directly proportional to the physical length l of the orbit. It is thuspossible to find periodic orbits by minimising the length by varying the reflection pointson the spheres.The length l can be expressed as a function of the 2nr angles ϕ1, ϑ1, . . . , ϕnr , ϑnr

describing the reflection points on the spheres. Using the reflection points as independentvariables, the gradient of the length function can be calculated. Finding periodic orbitsfor a given symbolic code has then been reduced to solving the equation

∇ l(ϕ1, ϑ1, . . . , ϕnr , ϑnr) = 0 , (4.29)

i.e. to a problem of multi-dimensional root search. This problem can be solved usingestablished numerical libraries, e.g. [32].

Once the orbit is known, the stability eigenvalues det(M − 1) can be computed. Fordetails on the calculation of the monodromy matrix, see [33, 34]. For this thesis, periodicorbit data from [8] has been used. Table 4.2 lists the first few periodic orbits in thefundamental domain as well as their properties for two ratios d/R. The table also givesthe maximum symmetry operation that leaves the orbit invariant. In the fundamentaldomain, this operation corresponds to the operation that maps the endpoint of the orbitin the fundamental domain onto the starting point. Note that, for example, the cycle t0which visits two spheres in turns is periodic in the fundamental domain, however, in fullphysical space, the orbit ends at the second sphere and thus is not periodic. Applicationof the rotation C2 respectively the reflection σd yields back the full periodic orbit.


Various methods to improve the convergence of the periodic orbit sum (4.6) have beenproposed. For systems with symbolic dynamics such as billiards, the method of cycleexpansion [35, 36] has proved to be especially successful; for an example see [37].

As preparation, we demonstrate how the periodic orbit sum (4.17) can be written as azeta function in the form of an Euler product over all periodic orbits

Z(k) =∏po

(1− tpo) , (4.30)

where tpo contains information on the periodic orbits’ properties. The explicit form ofthe dynamical zeta functions will be given for the case of 2 spatial dimensions. It willbe used to demonstrate the idea behind cycle expansions. First, it has been noted thatin 2 dimensions, there are two stability eigenvalues Λp,Λ

−1p , which may be written as

35


p hp lp Reλ1 Imλ1 Reλ2 Imλ2

0 σd, C2 0.50000 2.61803 0.00000 2.61803 0.000001 C3 0.76795 -3.48679 0.00000 2.99639 0.000002 S4 0.83542 -1.91725 2.80440 -1.91725 -2.8044001 σd 1.41753 -11.85590 0.00000 8.01328 0.0000002 C3 1.46422 -5.56213 8.54360 -5.56213 -8.5436012 S4 1.62677 11.46138 0.00000 -10.88920 0.00000

(a) d/R = 2.5

p hp lp Reλ1 Imλ1 Reλ2 Imλ2

0 σd, C2 2.00000 5.82843 0.00000 5.82843 0.000001 C3 2.26795 -7.09669 0.00000 5.75443 0.000002 S4 2.31059 -3.11111 5.69825 -3.11111 -5.6982501 σd 4.34722 -46.21054 0.00000 32.08725 0.0000002 C3 4.35831 -14.95013 35.68205 -14.95013 -35.6820512 S4 4.58593 43.79192 0.00000 -39.51750 0.00000

(b) d/R = 4

Table 4.2: Primitive periodic orbits up to cycle length np = 2 for (a) d/R = 2.5 and (b)d/R = 4. The reduced symbolic code p as well as the symmetry hp of each cycle isgiven. Furthermore, the real and imaginary parts of the stability eigenvalues λi aretabulated. All numbers have been rounded to five decimal digits. The shortest cycle,labelled by 0, which visits two spheres in turns has ambiguous symmetry. Both therotation about 180, C2, as well as the reflection about the plane perpendicular tothe line connecting the sphere’s centre, σd, map this particular orbit onto itself.

36


eup , e−up . We assume that all periodic orbits are very unstable so that u is large. Thestability prefactor then reads

|det(Mpr − 1)| =

(Λr + Λ−r − 2

)= eupr + e−upr − 2

= 2(cosh(upr)− 1)

= 4 sinh2(upr

2

). (4.31)

The square root appearing in (4.17) evaluates to√|det(Mp

r − 1)| = 2 sinh(upr

2

)≈ eupr/2 , (4.32)

where in the last step the assumption of large u has been used. Inserting (4.32) into(4.17), the semiclassical density of states reads

δgscl(k) =1

π

∑p

∞∑r=1

(−1)nprlp eirlpk−upr/2 , (4.33)

which is a geometric series in r. Summing up, we find

δgscl(k) =1

π

∑p

(−1)nplp exp [ilpk − up/2]

1− (−1)np exp [ilpk − up/2]. (4.34)

This expression may be written as

δgscl(k) = − 1

πIm

d

dklnZ(k) , (4.35)

with

Z(k) =∏p

[1− (−1)np exp (ilpk − up/2)] (4.36)

the dynamical zeta function. The product extends over all primitive cycles p. By thisreformulation, convergence of the periodic orbit sum is not yet improved. However, abetter convergence behaviour can be achieved by expanding the product [35, 36],

Z(k) = 1−∑q

Aq eilqk . (4.37)

The sum runs over all lengths lq, which themselves can be expressed in terms of periodicorbits,

lq =∑j

lpj . (4.38)

37


All information in phase and stability have been absorbed into the coefficients Aq. Thelq are called pseudo-orbits. Now, better convergence behaviour is achieved under theassumption that contributions from orbits respectively pseudo-orbits of the same lengthtend to cancel. This can be seen in the following example. Assume we consider a systemwith binary symbolic dynamics, e.g. the three-disk billiard. In this system, there are twofundamental cycles labelled by 0 and 1. Abbreviating

tp(k) := (−1)np exp (ilpk − up/2) (4.39)

in Z(k), the energies kn of the system can be calculated from the zeros of

Z(k) = (1− t0)(1− t1)(1− t10)(1− t100)(1− t110) · · · = 0

= 1− t0 − t1 − (t10 − t0t1)− (t100 − t10t0 + t110 − t1t10) . . . , (4.40)

where in the last step the product has been expanded and sorted by contributions fromcycles of different symbolic length. As can easily be seen, this infinite product convergesrapidly if the terms in parenthesis tend to cancel, e.g.

t10 ≈ t1t0 . (4.41)

This is the case under the conditions

l10 ≈ l1 + l0 ,

u10 ≈ u1 + u0 . (4.42)

This shadowing works specifically well for large configuration parameters d/R.The equations given above can be interpreted in such way that if shorter periodic orbits

act as building blocks for longer ones, the convergence behaviour of the semiclassicalapproximation is vastly improved. Note that in systems with smooth potentials, also theMaslov indices need to be additive, i.e.

σ10 ≈ σ1 + σ0 . (4.43)

This is trivially true in billiard systems.In (4.40), the shortest periodic orbits that define t0 and t1 give fundamental contribu-

tions. Longer cycles only provide corrections. For the shadowing to work, a completesymbolic dynamics is necessary, i.e. every symbolic code has to correspond to a realphysical orbit [36].Though the idea of cycle expansions was demonstrated for two-dimensional systems

with very unstable periodic orbits in which the approximation for the dynamical zetafunction holds, cycle expansions can be applied to more general systems as well. Insteadof the dynamical zeta function Zdyn, also the Gutzwiller-Voros zeta function ZGV canbe used. Also, the symbolic dynamics can be, for example, ternary instead of binary. A

38

4.4 Discrete symmetries and cycle expansion

cycle expansion of the Gutzwiller-Voros zeta function ZGV(k) given in (4.23) is achievedby inserting a bookkeeping variable z and thus rewriting

ZGV(k) = exp

(−∑p

∞∑r=1

1

r

(−z)rnp eirlpk√|det(Mp

r − 1)|

). (4.44)

This formula is cycle expanded if ZGV is expanded as a power series in z and thentruncated. The highest power of z equals the maximum cycle length nmax contributing tothe cycle expansion. After truncating, z has to be set to z = 1. We used cycle expansionup to order nmax = 14.


The method of cycle expansion allows to respect a system’s discrete symmetries. In suchsystems, the full physical spectrum can be decomposed into spectra belonging to differentrepresentations of the symmetry group. This decomposition will prove beneficial for thestudy of the fractal Weyl law (2.15).

4.4.1 The origin of degeneracy from symmetry

Below, the origin of degeneracies and the role of irreducible representations of thesymmetry group will be outlined following [38]. It is assumed that all degeneraciesoriginate from symmetry, i.e. there is no “accidental degeneracy”. The decomposition ofthe full spectrum into spectra belonging to a certain irreducible representation of thesymmetry group allows to study the spectra belonging to individual subspaces of the fullHilbert space.

Consider a system that has symmetry associated with an operator R. Application ofthe operator R on the coordinates describing the systems’ configuration maps the systemonto itself. In quantum mechanics, it is spoken of symmetry if the operator leaves theHamiltonian invariant. For example, the Coulomb problem has a continuous rotationalsymmetry, i.e. rotations of the coordinate system do not change the Hamiltonian; whereasthe four-sphere billiard has discrete tetrahedral symmetry. Consider the effect of R onthe coordinates r. The transformed coordinates r′ are defined as

r′ = Rr , (4.45)

or, in components

ri′ = Rijrj . (4.46)

The effect of the symmetry operation on the coordinates may thus be expressed as asimple matrix multiplication. The matrices for all symmetry operations Rk define theelements of a group with the usual matrix multiplication.

39


Now consider functions of the coordinates, e.g. the quantum mechanical wavefunctionψ = ψ(r). It is assumed there is an isomorphism mapping the coordinate transformationsR to operators which act on the wavefunction ψ. The effect of the symmetry operationon the wavefunction is given by an operator PR which is defined as

PR ψ(Rr) = ψ(r) . (4.47)

Alternatively, this can be stated as

PR ψ(r) = ψ(R−1r) . (4.48)

This definition preserves the group structure as required by the group axioms. To havegroup property,

PSPR = PSR (4.49)

has to hold for two transformations S and R. Applying PSPR to a wavefunction, wehave

PS [PR ψ(r)] = PS ψ(R−1r

)= ψ

(R−1

(S−1r

))= ψ

((SR)−1r

)= PSR ψ(r) . (4.50)

As this by definition holds for all wavefunctions ψ(r), (4.49) has been proved.Now consider all operators PR that commute with the Hamiltonian,

[PR, H] = 0 . (4.51)

These operators form the group of the Schrödinger equation. Applying such an operatorto the Schrödinger equation, we have

PRHψn = PREnψn

HPR ψn = EnPR ψn . (4.52)

Each transformed eigenfunction PR ψn thus is an eigenfunction degenerate to the originalone. Any gn-fold degenerate eigenvalue En is then associated with gn eigenfunctions.Applying any PR to one of these functions generates another wavefunction of the sameenergy. This new function can always be uniquely expressed as a linear combination ofthe complete set of the gn wavefunctions belonging to the energy En.1 This basis setspans a gn-dimensional subspace of the full Hilbert space H. All symmetry operationsof the Schrödinger group leave this subspace invariant. The symmetry operations map

1This assumes that there is no “accidental degeneracy” with origins different from symmetry.

40


any linear combination of the basis vectors onto another one. The group action canthus be understood from the action of the symmetry operations on the basis vectors.Considering a wavefunction ψv degenerate to gn eigenfunctions ψk, the action of PR onψv is expressed as

PR ψ(n)v =

gn∑k=1

ψ(n)k (Γ(n)(R))kv , (4.53)

where the matrix Γ(R)(n) is a representation2 of the transformation R. The index nindicates that there may be several independent representations of the transformation.From this definition of the representation Γ(R), it is evident that the dimension of therepresentation is equal to the multiplicity of the states belonging to the subspace underconsideration. As the identity operation E is always represented by the identity matrix,the multiplicity gn of states belonging to a certain subspace is given by

gn = tr Γ(E)

=: χ(E) . (4.54)

The character χ(E) can be read off from the symmetry group’s character table.The decomposition of the full spectrum containing all resonances into spectra containing

resonances belonging to a certain subspace allows us to study the fractal Weyl law (2.15)individually for each subspace. As the calculations for different subspaces differ inconvergence behaviour, advantage may be taken of these decompositions.

4.4.2 The symmetry group Td

The four-sphere billiard has discrete tetragonal symmetry. The associated symmetrygroup Td contains all symmetry operations that leave a regular tetrahedron invariant.In particular, there are the identity operation E, 4 rotations C3 by 2π/3 around theaxes defined by a vertex of the tetrahedron and the centre of the facing triangularboundary surface, 4 more rotations C3

2 by 4π/3 around the same axes, 3 rotations C2 byπ around the axes intersecting the middle points of opposing edges, 6 reflections σd atplanes perpendicular to the tetrahedron’s edges and also containing another vertex; and,furthermore, 3 permutations of the vertices S4 which can be written as a combinationof a rotation C4 by π/2 and a reflection σh at the plane perpendicular to the mainrotation axis, i.e. the axes of C3. Finally, the symmetry group Td also contains 3 distinctthree-times repeated rotary reflections S4

3.The character table of the symmetry group is given in table 4.3. The symmetry

can be decomposed into 5 invariant subspaces, i.e. the representation matrices D ofTd can be decomposed into block-diagonal form where the diagonal elements contain

2The group property Γ(SR) = Γ(S)Γ(R) can be shown from the definition.

41


Td E 8C3 3C2 6S4 6σd

A1 1 1 1 1 1A2 1 1 1 −1 −1E 2 −1 2 0 0T1 3 0 −1 1 −1T2 3 0 −1 −1 1

Table 4.3: Character table for the group Td. The group has five irreducible repre-sentations: the one-dimensional representations A1 and A2, the two-dimensionalrepresentation E and the two three-dimensional representations T1 and T2.

representations belonging to a certain representation. This representation is called anirreducible representation if no further decomposition is possible.From the character table, statements on the wavefunctions ψ can be made. For the

one-dimensional representations A1 and A2, the effect of the symmetry transformationsis described by a multiplication with the complex number given by the character χ. Forthe totally symmetric A1 subspace, all characters are equal to 1, so the wavefunctionshave the full symmetry of Td, i.e. ψ is not altered by any symmetry transformation. Inthe A2 subspace, the wavefunction changes sign under the reflection σd and permutationS4. For representations of higher dimension, the effect of the symmetry transformationsis not as simple to describe.

We note that repeated application of symmetry transforms may be identical with otherelements of the symmetry group, e.g. [39]

σd2 = E , (4.55a)

C22 = E , (4.55b)

C33 = E , (4.55c)

S42 = C2 , (4.55d)

S44 = E . (4.55e)

These identities will be useful for the symmetry decomposition of zeta functions demon-strated below.

4.4.3 Using symmetries in cycle expansions

Cycle expansions of zeta functions can account for the symmetries a system has. Following[40], it will be demonstrated below how discrete symmetries lead to symmetry factorisedzeta functions. These allow the computation of quantum spectra belonging to a specificsymmetry subspace.

42


In quantum mechanics, the full Hilbert space H of the problem factorises into subspacesbelonging to a certain irreducible representations of the symmetry group, i.e.

H = HA1 ⊗HA2 ⊗HE ⊗HT1 ⊗HT2 , (4.56)

for the four-sphere scattering system. Cvitanović and Eckhardt pointed out that zetafunctions can be factorised in a similar way [40]. The derivation given in the originalpublication is set in a quite general framework. Some key ideas will be briefly repeatedhere. First, the dynamical system under consideration is defined by a transfer operator

L = δ(y − f(x)) , (4.57)

which maps xn+1 = y = f(xn). The distribution f in our case is the quantum wavefunction; the transfer operator is the quantum propagator, which itself depends on theHamiltonian. The eigenvalues λk of L can be written as the inverse zeros of a zetafunctions

Z(z) = det(1− zL) =∞∏k=0

(1− zλk) . (4.58)

This expression may be evaluated in a periodic orbit framework. Using the relation

det(1− zL) = exp

(∞∑n=0

zn

ntrLn

)(4.59)

between an operator determinant and its trace, and a “periodic orbit” respectivelysemiclassical approximation, Cvitanović and Eckhardt arrive at the zeta function

Z(z) = exp

(∑p

∞∑r=1

1

r

znpr∣∣1− J rp

∣∣), (4.60)

with np the periods of primitive periodic cycles p, r the repeated revolutions and Jp thecycle Jacobians. This has essentially the same form as the zeta function (4.23).

Cvitanović and Eckhardt note that the fundamental domain of phase space is sufficientfor all computations, as the whole phase space M can be obtained from the fundamentaldomain M by

M =∑a∈G

aM , (4.61)

where G is the symmetry group. Resultingly, any starting point x can be mappedonto a point x inside the fundamental domain of phase space. Using this, Cvitanović

43


and Eckhardt evaluate the trace of the transfer operator appearing in (4.59) on thefundamental domain of q dimensional phase space to

trL =

∫M

ddxL(x,x)

=

∫M

dqx∑h

trD(h)L(h−1x, x) , (4.62)

where h is a symmetry operation and D(h) is a suitable representation thereof acting onthe transfer operator. Following [40, 36], this results in the identity

(1− tp)mp = det (1−D (hp) tp) , (4.63)

with mp the multiplicity of a primitive cycle p.These expressions may be evaluated using a certain explicit representation Dα(h).

However, this is rather time-consuming. Instead, the determinant may be expressed interms of traces that can be read from the symmetry group’s character table. First, notethat for a matrix A ∈ Rd×d with eigenvalues λi, we have

det(exp(A)) =d∏i=1

expλi . (4.64)

This can be understood assuming A = diag(λ1, ..., λd) is diagonal and then using theidentity [41]

A =

λ1 . . . 0... . . . ...0 . . . λd

=⇒ expA ≡

expλ1 . . . 0... . . . ...0 . . . expλd

. (4.65)

Also, note that

exp(tr(A)) = expd∑i=1

λi =d∏i=1

expλi . (4.66)

Combining (4.64) and (4.66), we conclude

det(exp(A)) = exp(tr(A)) . (4.67)

44


d det(1−D(h))t

1 1− χ(h)t

2 1− χ(h)t+ 12(χ(h)2 − χ(h2))t2

3 1− χ(h)t+ 12(χ(h)2 − χ(h2))t2 + 1

6(χ(h)3 − 3χ(h)χ(h2) + 2χ(h3))t3

Table 4.4: Expansions of det(1−D(h)) for different dimensions d of the representationmatrices D(h). The trace trD(h) is as usually denoted by χ(h). The tetrahedrongroup Td has two three-dimensional representations T1, T2. Therefore, expansions upto d = 3 are given.

Assume now the matrix A is given by A = 1 + εD(h) = exp(ln(1 + εD(h))). For ε = −1this specific choice of A is identical with the factors in cycle expansions. We may nowuse (4.67) and expand the logarithm and the matrix exponential into a power series:

det (1 + εD(h)) = exp (tr (ln (1 + εD)))

= exp

(tr

(−∞∑j=1

(−εD)j

j

))

= exp

(−∞∑j=1

(−ε)jj

tr(Dj)

)

=∞∑k=0

1

k!

(−∞∑j=1

(−ε)jj

tr(Dj)

)k

(4.68)

As there are only d eigenvalues, the result can at most be a polynomial of degree d in ε(compare (4.66)). Ignoring all terms in εj with j > d, expansions of the determinant fordifferent dimensionalities of the representation matrices may be obtained by truncatingthe series. Setting ε = −1, we can express the zeta functions factors for different valuesof d. Expansions up to d = 3 are compiled in table 4.4. Carrying out this procedureexplicitly, one can use equations (4.55) and (4.68) and obtains the factorisations given intable 4.5. Thus, we have demonstrated how the zeta function Z =

∏p(1 − tp) can be

rewritten in a symmetry reduced version

Zα =∏p

det(1−Dα(hp)tp) (4.69)

for the subspace α. The zeta function now depends only on the fundamental cycles p.By this procedure, a factorisation

Z(k) =∏α

Zα(k)dα (4.70)

45


E C3 C2 S4 σd

A1 (1− tp) (1− tp) (1− tp) (1− tp) (1− tp)A2 (1− tp) (1− tp) (1− tp) (1 + tp) (1 + tp)E (1− tp)2 (1 + tp + t2p) (1− tp)2 (1− tp)(1 + tp) (1− tp)(1 + tp)T1 (1− tp)3 (1− tp)(1 + tp + t2p) (1− tp)(1 + tp)

2 (1− tp)(1 + t2p) (1− tp)(1 + tp)2

T2 (1− tp)3 (1− tp)(1 + tp + t2p) (1− tp)(1 + tp)2 (1 + tp)(1 + t2p) (1 + tp)(1− tp)2

Table 4.5: Symmetry factorisation of the zeta function Z for all five irreducible represen-tations of the group Td. The table entries give the contribution of each fundamentalcycle p to the Euler product Z =

∏p(1−tp). This factorisation allows the computation

of quantum spectra for each symmetry subspace.

is achieved. The zeta function Z factorises into zeta functions belonging to certainirreducible representations α of the symmetry group. The dimensions dα of the represen-tations enter into the full zeta function – and with them, the quantum multiplicities ofresonances belonging to a certain subspace.

Assigning weight factors

The method of cycle expansion demonstrated in section 4.3 expands the zeta function Zinto a truncated series in which all cycles up to a certain cutoff length enter. However,besides the primitive cycles, also multiple traversals contribute. Therefore, it needs tobe clarified how repeated revolutions can be taken into account. Presume the primitivefundamental cycles p are known. Then, the contribution of an r-times repeated revolutionto the symmetry reduced zeta function (4.69) is given by polynomials such as

(1− zrtp,r) , (4.71)

where a dummy variable z has been introduced. The cycle weights tp,r have the form ofthe summands in (4.23) and are thus easily calculable from the cycle weight tp of theprimitive fundamental cycle. By using the factorisations given in table 4.5, it is possibleto determine the weight factor wp,r as the sum of all roots zir of the polynomials givenin the table,

wp,r =∑i

zir . (4.72)

If this is possible, a way to use the p for repeated revolutions as well has been found. Asan example, for the contribution of the r-times repeated cycle p to the A1 spectrum, weneed to solve

0 = (1− zrtp,r) , (4.73)

46


E C3 C2 S4 σd

A1 1 1 1 1 1

A2 1 1 1 (−1)r (−1)r

E 2 2 cos(

2π3r)

2 1 + (−1)r 1 + (−1)r

T1 3 1 + 2 cos(

2π3r)

1 + 2(−1)r 1 + 2 cos(π2r)

1 + 2(−1)r

T2 3 1 + 2 cos(

2π3r)

1 + 2(−1)r (−1)r + 2 cos(π2r)

2 + (−1)r

Table 4.6: Weight factors wp,r for r traversals of the primitive cycle p. These factorsallow for symmetry factorisations with repeated revolutions of primitive fundamentalcycles.

which is true for zr = 1. Thus, in the A1 subspace, all weight factors wp,r are 1. Bythis choice, the symmetry factorisation is retained. As another example, consider the Esubspace for cycles with C3 symmetry. Here, solutions to the equation

0 = (1 + zrtp,r + zrtp,r2) (4.74)

are needed. A factorisation is given by

(1− e2πir/3 tp,r

) (1− e−2πir/3 tp,r

)= 0 , (4.75)

where the exponentials are the roots zi. Evaluating the sum z1r + z2

r, we find the weightfactors wp,r = −1,−1, 2,−1,−1 . . . for r = 1, 2, . . . . A short notation for this sequenceis given by wp,r = 2 cos(2πr/3). By similar calculations, the weight factors wp,r given intable 4.6 are determined.

Ambiguous symmetry

The shortest cycle labelled by 0 in the four-sphere system has ambiguous symmetry. Itis possible to map this cycle onto itself by both the rotation C2 and the reflection σd.This ambiguity requires special care in the symmetry decomposition. This is particularlyimportant as the 0-cycle is one of the fundamental cycles that act as building block forlonger cycles in the sense of cycle expansion. The group theoretical weight of the 0-cyclecan be written as [40]

h0 =C2 + σd

2. (4.76)

47


The symmetry factorisation can thus be not one of those given in table 4.5. However, itis possible to use a factorisation that contains factors in such a way that the factorisationis at most the greatest common divisor of the factors given in table 4.5, which are

A1 : (1− t0)

A2 : 1

E : (1− t0)

T1 : (1− t0)(1 + t0)2

T2 : (1 + t0)(1− t0)

A possibility to check which one is correct, the full zeta function Z(k) =∏

α Zα(k) canbe calculated as the product of the zeta functions Zα of all irreducible representationsα. The result can be examined for the contribution of the 0-cycle. As the 0-cycle is notperiodic in full physical space, but rather the twice repeated revolution, only terms witht20 are allowed to appear. Furthermore, as there are 6 =

(42

)possibilities of visiting two

spheres in turns, we require a coefficient of 6, i.e. 6t20 has to appear. We find this to betrue for the factorisations

A1 : (1− t0)

A2 : 1

E : (1− t0)

T1 : (1 + t0)

T2 : (1 + t0)(1− t0) .

The expanded result sorted by cycle length then reads

Z(k) = 1 + 0t0 − 6t20 − 8t31 + 3(5t40 − 4t201 − 2t42) . . . . (4.77)

Note that the 1-cycle, which corresponds to visitations at three different spheres in aplane, is only periodic in full physical space after 3 revolutions; furthermore, there are8 = 2

(43

)distinct sequences of length 3 constructable from 4 labels. Thus, the terms 8t31

is required to appear in the non-symmetry reduced zeta function. Similar combinatorialarguments can also be made for the cycle labelled by 2. In conclusion, we find (4.77)to be the correct zeta function. The weight factors w0,r calculated from the expansionsgiven above are given in table 4.7. The explicit form of the zeta function derived from(4.23) we use for our calculations is

ZGV;α(k) = exp

(−∑p

∞∑r=1

1

r

wp,r;α(−z)rnp eirlpk√|det(Mp

r − 1)|

), (4.78)

with p the primitive symmetry reduced cycles, r the repeated revolutions thereof andα the symmetry subspace. From this equation, a symmetry reduced version of cycle

48

4.5 Harmonic inversion method

α A1 A2 E T1 T2

w0,r 1 0 1 (−1)r 1 + (−1)r

Table 4.7: Weight factors w0,r for the fundamental cycle 0 in all subspaces α of thesymmetry group Td.

expansion is obtained by expanding (4.78) into a power series in z which is truncated ata maximum cycle length nmax. Then, z has to be set to z = 1. With these preparations,quantum spectra can be computed. The energy eigenvalues kn can be obtained as thecomplex zeros of Z(k). Details on an alternative to numerical root search are given inthe next section.


As shown above, the zeta function Z(k) contains all energy eigenvalues k as complexzeros. It is possible to obtain spectra by root search. This method has been successfullyused for billiards; compare e.g. [37]. However, root search in cycle expansions of highorder is numerically expensive. For statistical purposes it is important not to miss anyresonances in the strip of the complex plane under consideration. Therefore, a dense gridof initial root guesses has to be used for the root search. Consequently, many resonanceswill be found several times. Thus, the problem is to distinguish for every new rootwhether a new distinct resonance has been found or if the new zero has already beencomputed. As the number of resonances enters into the fractal Weyl law (2.15) throughthe counting functions N(k), it is crucial to count individual resonances only once.An alternative to the computation of zeros is the harmonic inversion method for

high-resolution spectral analysis first introduced by Wall and Neuhauser [42] and laterimproved by Mandelshtam, Taylor, Main and others. The harmonic inversion method candirectly be used on periodic orbit data to find quantum eigenvalues. For an overview ofapplications in semiclassical physics, see [43]. For this thesis, however, harmonic inversionwill solely be used for spectral analysis. The harmonic inversion variant discussed herewas developed by Belkić, Dando, Main and Taylor [44, 14]. The problem of extractingeigenvalues is restated as a signal processing task. The explanation given below is basedon [14].

4.5.1 Harmonic inversion technique

In section 4.1.1 it has been shown that the density of states g(k) relates to the trace ofthe Green’s function

G(k) =∑n

1

k − kn, (4.79)

49


whose poles correspond to energy eigenvalues kn. In the semiclassical limit, the densityof states can be expressed in terms of the periodic orbit sum

gscl(k) =∑p

∑r

Ap,r eiklp,r , (4.80)

by using the identity

g(k) = − 1

π

∑n

Im1

k − kn. (4.81)

Another formulation had been found in the Euler product

Z(k) =∏p

(1− tp) (4.82)

which can be used for cycle expansion; compare section 4.3. Presume the cycle expansionof g(k) is evaluated along the real axis k ∈ R. With the complex energies kn = kn− iΓn/2,a real valued signal

g(k) = − 1

πIm∑n

1(k − kn

)+ iΓn

2

= − 1

πIm∑n

(k − kn

)− iΓn

2(k − kn

)2+(

Γn2

)2

=∑n

1

π

Γn2(

k − kn)2

+(

Γn2

)2 (4.83)

is obtained. This signal is a superposition Lorentzians centred at the positions knwith widths Γn. The Lorentzians can overlap strongly as illustrated in figure 4.1. Thechallenge of extracting the individual parameters of the Lorentzians, i.e. positions, widths,amplitudes from the real-valued signal g(k) is tackled by the harmonic inversion technique.

Assume the superposition of Lorentzians to fit is given by the general formula

g(k) =∑n

An(k − kn

)2+(

Γn2

)2 , (4.84)

where we additionally allow for amplitudes An. Again, the kn are complex values

kn = kn − iΓn2. (4.85)

We first note that the k-space signal (4.84) can be Fourier transformed to position space;it then reads

C(x) := F(g(k)) =∑n

dn e−iknx , (4.86)

50


10 20 30 40 50 60k

0

5

10

15

20

25

30

35

40g(k)

Figure 4.1: Spectrum g(k) computed for d/R = 6. The spectrum is a superpositionof strongly overlapping Lorentzians; compare (4.84). The positions, widths andamplitudes of the Lorentz profiles can be determined by high-resolution harmonicinversion.

where the complex numbers dn are related to the amplitudes An.Then, as the first step of harmonic inversion, the real signal g(k) – discretely sampled at

a sampling interval ∆k – is transformed to position space by a discrete Fourier transform(DFT). Furthermore, the DFT signal is band-limited, i.e. multiplied by a window functionin x-domain or convoluted with the window function in k-domain. To limit the numbersof frequencies in the DFT, the window function generally has to decay to 0 as k goes to±∞. The simplest possibility to accomplish this is to use a rectangular window function,for which the windowing process can be reduced to a simple choice of suitable summationbounds in the DFT. With a window interval of [k0 −∆k/2; k0 + ∆k/2], the windowedDFT leads to the band-limited (BL) discrete signal

cn := CBL(x = j∆x) =

m2∑m=m1

g(km) ei(km−k0)j∆x , (4.87)

with equidistant samples separated by ∆x = 2π/∆k. The summation bounds m1

and m2 have to be chosen such that the grid points km are contained in the interval[k0 −∆k/2; k0 + ∆k/2]. The “frequencies” km of the oscillating exponential in (4.87)

51


are shifted by k0 in order to reduce phase oscillations. In equation (4.87), the DFT isdefined on a discrete set of points j∆x with j = 0, . . . NBL−1. The number of supportingpoints in that window of the spectrum (4.84), which is to analyse, limits the reasonablechoice of NBL to NBL ≤ m2−m1 + 1. Choosing a higher number of supporting points, noadditional information is contained in the DFT. We thus set NBL = 2N . The number Nwill be shown to be an upper bound for the number of resonances that can be resolved.

The second step of harmonic inversion is to solve (4.87) for the “frequencies” k′n = kn−k0

in the Fourier transform (4.86), i.e. to solve

cj =N∑n=1

dnzjn ; j = 0, . . . , 2N − 1 , (4.88)

with the DFT kernel zn = eik′n∆x. For small numbers of “frequencies”, N ≈ 100, a methodbased on Padé approximants has proved to be successful [44, 14]. First it is assumedthat the signal cj has infinite length, i.e. j = 0, . . . ,∞. Then, a function f(z) can beconstructed from

f(z) :=∞∑j=0

cjz−j , (4.89)

which is a Taylor series about 0 in the variable z−1. Inserting (4.88), the defining equationfor f(z) reads

f(z) =N∑n=1

dn

∞∑j=0

(znz

)n. (4.90)

Using the geometric series∞∑j=0

(znz

)j=

z

z − zn, (4.91)

f(z) can be written as

f(z) =N∑n

dnz

z − zn. (4.92)

This result is exact provided the geometric series is absolutely convergent, i.e. zn/z < 1 forall n = 1, . . . , N . Furthermore, for exactness, the series

∑j cjz

−j has to exist for all valuesz. However, if these conditions are not met, (4.92) provides an analytic continuation to(4.89). Equation (4.92) constitutes a polynomial approximation to (4.89); f(z) may bewritten as

f(z) =PN(z)

QN(z), (4.93)

52


where both PN(z) and QN(z) are polynomials of degree N in the variable z; thus, f(z)is a rational function which has single poles at the roots of the denominator polynomialQN(z). Using this ansatz, the quantities dk can be calculated by residue calculus. If thecomplex contour integral∮

γ

f(z) dz =

∮γ

N∑n

dnz

z − zndz =

∮γ

PN(z)

QN(z)dz (4.94)

over a closed curve γ that encloses the pole at z = zn is evaluated, we find

2πi dnzn = 2πiPN(zn)dQNdz

∣∣zn

(4.95)

from the last two terms in (4.94). Solving for dn gives

dn =PN(zn)

zndQNdz

∣∣zn

. (4.96)

Substituting dn back into (4.88) then allows to solve the equation for the frequencies.However, as the signal cj is of finite length, this procedure cannot be carried out inpractise. Yet, using Padé approximants, the signal points cj with j = 0, . . . , 2N − 1 aresufficient to determine the coefficients of the two polynomials

PN(z) =N∑n=1

bnzn (4.97)

and

QN(z) =N∑n=1

anzn − 1 . (4.98)

These two polynomials form the rational function (4.93). By choosing the coefficientssuch that the rational function yields the same function value as (4.92) at z = 0,f(0) = PN(0)/QN(0), and, additionally, such that the i -th derivative (i = 1, . . . , N)agrees, then the quotient PN(z)/QN(z) is called a Padé approximant to f(z) [45, 46].Padé approximants are typically used if the function f(z) is not known analytically.

As PN (0) = 0, the condition PN (0)/QN (0) = f(z = 0) = 0 is met trivially. By equatingthe rational function (4.93) with (4.89) in finite summation bounds 0 and 2N − 1 andmultiplying both sides with QN(z), the equation

2N−1∑i=0

cizi

(N∑n=1

anzn − 1

)=

N∑n=1

bnzn (4.99)

53


is obtained. Comparing like powers in z, the linear set of equations

ci =N∑i=1

anci+n , i = 0, . . . , N − 1 (4.100)

for an is obtained. Using the solution thereof, the bn are found to be

bn =N−n∑m=1

an+mcm , n = 1, . . . , N . (4.101)

Finally, the parameters zn = exp(−ik′n∆x) respectively the “frequencies”

kn = k0 + k′n = k0 +i

∆xln zn (4.102)

are computed by a root search for the polynomial QN(z). This can be established as alinear algebra problem. First, construct an upper Hessenberg matrix

A =

−aN−1/aN −aN−2/aN . . . −a1/aN −a0/aN

1 0 . . . 0 00 1 . . . 0 0...

......

...0 0 . . . 1 0

. (4.103)

The matrix A corresponds to the polynomial QN(z), as the characteristic polynomialχ(z) is equivalent to QN(z),

χ(z) = det(A− z1) = QN(z) . (4.104)

The eigenvalues of A solve

χ(z) = 0 , (4.105)

and are thus the zeros of QN(z). The “frequencies” kn can be obtained by diagonalisingA, which is a standard numerical problem [47]. The amplitudes of individual resonancesare given by (4.96). The amplitudes can be used to distinguish correct resonances andwrong ones. This will be discussed in subsection 4.5.3.

A sample spectrum obtained from harmonic inversion is shown in figure 4.2. In someregions of the plot, resonances tend to be located close to the real axis. However, onemight suspect further resonances deeper in the complex plane. For eigenvalue statistics, itis crucial to find as many resonances as possible. It is thus desirable to investigate largerregions of the complex plane. A method to resolve more resonances is demonstrated inthe following subsection.

54


0 200 400 600 800 1000Re k

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

Imk

A1 resonances

Figure 4.2: Sample output of harmonic inversion for d/R = 6. It can be seen that forlarge real parts Re k the resonances are close to the real axis. Resonances deeper inthe complex plane are not resolved.

4.5.2 Investigating regions deeper in the complex plane

Harmonic inversion cannot resolve very broad Lorentzians. The method described belowsharpens the Lorentzians’ peaks in such a way that an investigation of regions deeper inthe complex plane is possible.If the spectrum (4.83) is not constructed from an evaluation of g(k) along the real

axis, but rather along the shifted line k + iδ with real k and δ, the spectrum (4.83) reads

g(k) =∑n

1

π

Γn+2δ2(

k − kn)2

+(

Γn+2δ2

)2 (4.106)

This spectrum contains Lorentzians at the same centre positions kn, but, if a negative δis chosen, with smaller widths Γn + 2δ. This approach is illustrated in figure 4.3. Theharmonic inversion output of a shifted and an unshifted spectrum are shown in figure 4.4.Using different shifts, a complete spectrum can be composed from calculations withdifferent parameters δ. Each calculation yields precise resonances in a certain strip ofthe complex plane. Filtering resonances outside the individual strips and adding strips,the complete spectrum can be constructed.

55


0 5 10 15 20 25 30 35 40k

0

1

2

3

4

5

6

g(k)

δ=0

δ=−0.3

Figure 4.3: Comparison of two A1 spectra g(k) for d/R = 6. The spectrum drawnin red was evaluated along the real axis, whereas the spectrum drawn in blue wasshifted by δ = −0.3. It is evident from the plot that the peaks in the spectrum forδ = −0.3 are sharper. The overlap of the Lorentzians is reduced and thus, harmonicinversion is eased. Note that for clarity, the red curve was shifted and stretched.

4.5.3 Procedure

In conclusion, the procedure of calculating quantum spectra used for this thesis is asfollows:First, the spectrum g(k) is calculated as a superposition of Lorentzians. We use the

cycle expanded zeta function Z(k) for this purpose. The quantity

g(k) =d

dklnZ(k) =

Z ′(k + iδ)

Z(k + iδ)(4.107)

is evaluated along lines parallel to the real axis with different shifts δ. Thus, the shiftsthat allow for better results in harmonic inversion enter into the cycle expansion. Then,harmonic inversion is used to obtain the Lorentzians’ parameters kn and Γn for spectracalculated with different shifts. In the next step, the spectra are filtered via the complexamplitudes. The quantity g(k) given in (4.107) should give resonances with an amplitudeof An = i. We allow for amplitudes An located in circle around z = i in the complexplane with a radius of R0. We used R0 = 0.5 for our calculations. This filtering separates

56


0 100 200 300 400 500 600 700Re k

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

Imk

δ = −0.3

δ = −0.45

Figure 4.4: Comparison of harmonic inversion output from two calculations for thesubspace A1 at d/R = 6 with shifts δ = −0.3 (drawn in red) and δ = −0.45 (drawnin blue). Calculations with different shifts can be used to construct a spectrum thatcontains all resonances that can be obtained from finite length cycle expansions.

correct resonances from spurious ones. The filtering is illustrated in figure 4.5. Finally,the spectra for different shifts δ are joined such that the individual strips do not overlap.

57


0 50 100 150 200 250 300Re k

−0.34

−0.32

−0.30

−0.28

−0.26

−0.24

−0.22

Imk

0.15

0.30

0.45

0.60

0.75

0.90

1.05

Figure 4.5: A1 resonance spectra for d/R = 10 calculated along the real axis. Theabsolute values of the amplitudes are mapped onto the colour in which the resonancesare drawn. The amplitudes are almost “binary”. Correct resonances have an amplitudeof An ≈ 1, whereas wrong resonances have amplitude An ≈ 0.

58

5 Results

The fractal Weyl law has been put to test for billiard systems before. In [4], the 3-diskbilliard has been studied. To make our own results comparable to those given therein,we aim at a discussion similar to the one given in [4].

For this thesis, the four-sphere scattering system has been studied for various valuesof the configuration parameter d/R. Furthermore, spectra belonging to the differentsubspaces of the symmetry group have been computed. In this section, resonance spectrawill be given; their features will be discussed briefly and characteristics of the spectraimportant for the counting function (2.10) will be pointed out. Then, counting will beperformed. The resulting functions will be used to determine exponents for the fractalWeyl law (2.15). The results will be compared with the exponents obtained in chapter 3,thus, the fractal Weyl law will be put to test.

5.1 Defining a scale for the strip widths

For the 3-disk system discussed in [4], the strip widths C have been chosen in relation tothe classical escape rate γ0. For large energies k →∞, the imaginary part of quantumresonances converges to Im k = −γ0/2 [4, 48]. Thus, the discussion of the results issimplified by rescaling the strip widths C to

C :=Cγ0/2

, (5.1)

which defines a universal scale independent of the symmetry subspace and the ratio d/R.Similar to [4], we evaluate the fractal Weyl law for scaled strip widths C ∈ [1; 1.6].The classical escape rate γ0 can be interpreted descriptively as follows [49]: presume

the scattering system under consideration is located in a box much larger than thesystem itself. Conducting N0 scattering experiments with the same incident energy k,but different incident directions, one finds that the number Nt of trajectories that areinside the box after the time t has passed decays as an exponential law

Nt ∝ N0 e−γ0t , (5.2)

where generally both the constant of proportionality and the escape rate γ0 depend onthe energy k. Scaling properties of billiard systems allow to choose the energy as k = 1.The relation of the escape rate and the imaginary part of the quantum resonances can

59

5 Results

be understood from the correspondence principle. The number of classical trajectoriesinside the box corresponds to the quantum probability density 〈ψ|ψ〉. The decay of thisprobability,

〈ψ|ψ〉 ∝ e−Γt , (5.3)

relates to the decay of the number of classical trajectories inside the box given by (5.2).Thus, in the classical limit

Im kn = −Γn2

class.−→limit−γ0

2(5.4)

holds.

5.1.1 Calculating the escape rate

The escape rate can be calculated by the method of cycle expansion as well [50]. Theescape rate γ0 is found to be the largest real zero of a dynamical zeta function

Z(s) =∏p

(1− tp) , (5.5)

with p the primitive periodic cycles and tp the cycle weights. For a three-dimensionalsystem,

tp =e−lps∣∣∣Λ(1)p Λ

(2)p

∣∣∣ . (5.6)

The quantities Λ(i)p are the leading stability eigenvalues. A generalisation to a Gutzwiller-

Voros type zeta function for a three-dimensional system is given by

Z(s) = exp

−∑p

∑r

1

r

e−rlps∣∣∣Λ(1)p Λ

(2)p

∣∣∣r , (5.7)

with Λ(i)p the leading two stability eigenvalues of the primitive cycle p and r the repeated

revolutions of the primitive cycle p. This zeta function can be cycle expanded as describedin section 4.3. Figure 5.1 shows Z(s) for d/R = 4. Results for the escape rate γ0 atvarious configurations d/R are given in table 5.1.

The tendency of γ0 to decrease as d/R increases can be understood from the factthat the travelling time of a particle between two bounces increases as d/R increases.Scattering events that may lead to a particle’s escape from the system become rarer asthe travelling time between the spheres increases.

60

5.2 Resonance spectra

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

1.16 1.162 1.164 1.166 1.168 1.17 1.172 1.174

Z(s

)

s

O(1)O(2)O(3)

Figure 5.1: Cycle expanded Gutzwiller-Voros type zeta function Z(s) evaluated ford/R = 4 and real s in the lowest three orders of cycle expansion. The zero gives thevalue of the escape rate γ0 .


We have computed spectra for all five distinct subspaces A1, A2, E, T1 and T2 at variousseparations d/R. In this section, we will compare spectra in terms of convergence.

5.2.1 Comparing different orders of cycle expansion in differentsubspaces

In cycle expansions of different order, we may consider resonances as converged withina small error if two successive orders of cycle expansion yield the same resonances. Infigure 5.2, a comparison of A1 resonances for d/R = 6 in orders 12 and 13 is shown.

Due to computation time and numerical precision, cycle expansions are practically onlymanageable for length up to about 12 or 13, so we only have limited choice in convergedresonances to work with.

As a general tendency, we noticed that the region of convergence in the complex planedecreases in size as subspaces of increasing dimension are considered. Thus, the A1

resonances generally provide good convergence behaviour. For comparison, T2 spectra

61

5 Results

γ0 O(1) O(2) O(3) O(4)

d/R = 4 1.16655 1.16459 1.16440 1.16440d/R = 6 0.85042 0.84977 0.84974 0.84974d/R = 8 0.68259 0.68230 0.68230 0.68230d/R = 10 0.57634 0.57619 0.57619 0.57619

Table 5.1: Classical escape rates γ0 in various orders of cycle expansion and variousvalues of the configuration parameter d/R.

are shown in figure 5.3. The A1 resonances are ideally suited for cycle expansion as inA1 all weight factors wp are equal to 1. In the other subspaces, the shadowing worksless well as different symmetry classes of the periodic orbits lead to different weightfactors. Compare e.g. the orbits labelled by 1 and 2 which have the symmetry classesC3 and S4. For the cycle expansion to be the most effective, the orbit labelled by 12which has symmetry class S4, should now approximately cancel in an expression such asw1w2t1t2 − w12t12. This is not the case for different weight factors wp.

Since cycle expansion of zeta functions is not an expansion that is absolutely convergentin a mathematical sense, it might be reasonable to work with cycle expansions of fairlylow order. However, we have noted that while small maximum cycle lengths of about 6or 7 usually allow to compute resonances deep in the complex plane for small energies ofabout Re k . 200, an investigation of resonances with large energies requires much highercycle lengths. Similar behaviour was already encountered for the three-disk billiard [37].Another important observation is that large configuration parameters d/R meets theprerequisites of cycle expansions much better, thus, lower orders may be used in thecalculations. We used a maximum cycle length of 11 for d/R = 10, whereas for d/R < 10,a maximum cycle length of 13 was used.

5.2.2 Spectra

Similar to the discussion given in [4], we used rescaled strip widths C = 2C/γ0 ∈ [1; 1.6].The spectra used in our discussion are given in this subsection.

Spectra for d/R = 4

For small configuration parameters such as d/R = 4, cycle expansion does not workvery well. This is illustrated in figure 5.4. Due to the very limited number of convergedresonances, we pass on these spectra for a discussion of the fractal Weyl law.

62


Spectra for d/R = 6

For d/R = 6, cycle expansion provides better results than for the smaller value of d/R = 4.The spectra belonging to the one-dimensional group representations A1 and A2 haveconverged well. Spectra for these two symmetry subspaces are shown in figure 5.5Cycle expansions for higher dimensional subspaces E, T1, T2 yield less converged

resonances. To illustrate this problem, a comparison of spectra computed from twoconsecutive orders of cycle expansion is shown in figure 5.6. For reasons of convergence,we pass on these spectra for d/R = 6.

Spectra for d/R = 8

At the intermediate value of d/R = 8, we have found the A1 and A2 to be converged forRe k < 3000. The spectra are shown in figure 5.7

In all other subspaces, we found the amount of converged resonances to too small. Forthis reason, these spectra will be not used in an evaluation of the fractal Weyl law.

Spectra for d/R = 10

At a configuration of d/R = 10, the best convergence behaviour is to be expected. TheA1 and A2 resonances shown in 5.8 have found to be converged.In the E subspace, resonances with real parts up Re k = 1500 have found to be

converged in the strip 2C/γ0 ∈ [1; 1.6]. The spectrum is shown in figure 5.9. However, asa very large k-range is desirable for a proper test of the fractal Weyl law, we will onlyuse the A1 and A2 spectra.

63

5 Results

-0.7

-0.65

-0.6

-0.55

-0.5

-0.45

-0.4

-0.35

-0.3

0 200 400 600 800 1000

Im k

Re k

O(12)O(13)

(a)

-0.7

-0.65

-0.6

-0.55

-0.5

-0.45

-0.4

-0.35

-0.3

1000 1100 1200 1300 1400 1500 1600 1700 1800 1900

Im k

Re k

O(12)O(13)

(b)

Figure 5.2: Comparison of A1 resonances for order 12 of cycle expansion (red) andorder 13 (green) for d/R = 6. For the range of Re k drawn in (a), the resonances areconverged, whereas in (b) only resonances in the strip between the real axis and theIm k = −0.55 have converged. This imaginary part corresponds to a rescaled stripwidth C = 1.29.

64


-0.7

-0.65

-0.6

-0.55

-0.5

-0.45

-0.4

-0.35

-0.3

0 100 200 300 400 500

Im k

Re k

O(12)O(13)

Figure 5.3: Comparison of T2 resonances for order 12 of cycle expansion (red) and order13 (green) at d/R = 6. Clearly, convergence is much worse than for the subspace A1;compare figure 5.2.

65

5 Results

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

0 200 400 600 800 1000

Im k

Re k

-γ0/2

O(12)O(13)

Figure 5.4: A1 resonances for the configuration d/R = 4 obtained from cycle expansionin orders 12 and 13. The blue line indicates at Im k = −0.58220 = −γ0/2 indicatesthe asymptotic imaginary part of the resonances in the classical limit. As can clearlybe seen, only a small amount of resonances is converged.

66


-0.65

-0.6

-0.55

-0.5

-0.45

-0.4

-0.35

-0.3

0 500 1000 1500 2000

Im k

Re k

-γ0/2

(a)

-0.65

-0.6

-0.55

-0.5

-0.45

-0.4

-0.35

0 500 1000 1500 2000

Im k

Re k

-γ0/2

(b)

Figure 5.5: Resonance spectra calculated from cycle expansions with a maximum cyclelength of 13 for the configuration d/R = 6 in (a) the A1 subspace and (b) the A2

subspace. Whereas the A1 resonances have converged very well, the A2 resonanceswith real parts Re k > 1500 and imaginary parts Im k < −0.535 are not converged.This has to be taken into account for a test of the fractal Weyl law in this subspace.

67

5 Results

-0.65

-0.6

-0.55

-0.5

-0.45

-0.4

-0.35

800 1000 1200 1400 1600 1800

Im k

Re k

-γ0/2

O(12)O(13)

Figure 5.6: Comparison of parts of E spectra computed from cycle expansions of length12 and 13 for the configuration d/R = 6. The convergence problem is clearly visible.

68


-0.5

-0.45

-0.4

-0.35

-0.3

-0.25

0 500 1000 1500 2000 2500 3000

Im k

Re k

-γ0/2

(a)

-0.5

-0.45

-0.4

-0.35

-0.3

0 500 1000 1500 2000 2500 3000

Im k

Re k

-γ0/2

(b)

Figure 5.7: Resonance spectra calculated from cycle expansions with a maximum cyclelength of 13 for the configuration d/R = 8 in (a) the A1 subspace and (b) the A2

subspace.

69

5 Results

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

0 1000 2000 3000 4000 5000

Im k

Re k

-γ0/2

(a)

-0.45

-0.4

-0.35

-0.3

-0.25

0 1000 2000 3000 4000 5000

Im k

Re k

-γ0/2

(b)

Figure 5.8: Resonance spectra for the configuration d/R = 10 in order 11 of cycleexpansion for (a) the A1 subspace and (b) the A2 subspace.

70


-0.45

-0.4

-0.35

-0.3

-0.25

0 200 400 600 800 1000 1200 1400

Im k

Re k

γ0/2

Figure 5.9: E resonances in order 11 of cycle expansion for d/R = 10.

71

5 Results

1

10

100

1000

10000

100000

1e+06

1 10 100 1000 10000

N(k

)

k

C = 0.28

C = 0.34

C = 0.35

C = 0.36

C = 0.37

C = 0.45

const⋅x1.1888

Figure 5.10: Counting functions N(k) for A1 resonances calculated from cycle expansionin order 11 for the ratio d/R = 10. Several counting functions for different stripsC are shown. The solid black line is the power law expected from the fractal Weyllaw (2.15). The deviations of the counting functions from straight lines are discussedin the text. The curves can be used as “raw data” to fit power functions to. Thisway, the exponent α in the fractal Weyl law can be obtained and compared to theclassical calculations.

5.3 Counting resonances

Spectra such as those shown in the preceding section can be used to count resonances.In this section, counting functions will be shown. From those, exponents for the fractalWeyl law will be given and compared with the exponents obtained from the classicalcalculations.The fractal Weyl law (2.15) suggests that the counting functions N(k) obey a power

law,

N(k) ∝ kα , (5.8)

thus, in a logarithmic plot of N(k), straight lines of slope α are expected. Samplecounting functions for d/R = 10 are plotted in figure 5.10. This figure is quite generic in

72

5.4 Putting the fractal Weyl law to test

structure, i.e. we have found similar features of N(k) in other subspaces and for otherratios d/R as well. Thus, a brief discussion of these features will be given in the following.We first note that the strip width C has to be sufficiently large, as in any other case,

counting would not involve reasonably large numbers of resonances. In the specificspectrum used in the construction of the counting functions, we have found goodresonances in the relevant strip C ∈ [1; 1.6] for Re k . 6000. For small strip widthssuch as C = 0.28⇔ C = 0.97, it is evident from figure 5.10 that counting involves onlya limited number of resonances. Larger strip widths involve more resonances in thecounting. However, choosing the strip width too large, the counting may also involveresonances that may not have been converged. Taking the asymptotic behaviour of theresonances’ imaginary parts into account, choosing rescaled strip widths in the intervalC ∈ [1.0; 1.6] seems to be a reasonable choice.The figure 5.10 reveals that the counting functions N(k) are not exactly given by

power laws. Power laws led to straight lines in the plot. From this observation one infersthat the exponent α will clearly depend on the k-range one fits to. We follow [4] andchoose the largest interval converged resonances have been computed for. Furthermore,we will demonstrate the effect of choosing different intervals.

5.4 Putting the fractal Weyl law to test

To provide as many tests for the fractal Weyl law as possible, we will provide plotsshowing the exponents α obtained from least-squares fitting a power function N(k) ∝ kα

to the counting functions calculated from the spectra for various subspaces, fitting ranges[kmin; kmax], configuration parameters d/R and strip widths C. We have performed least-squares fits to match the function (5.8) to the measured N(k). Another possibility wouldbe to fit to logarithm logN(k) = α log k+ const. However, this procedure corresponds toa specific but unusual choice of an error norm. We choose the more “natural” possibilityto use the method described first.The results for the three-disk billiard obtained by Lu, Sridhar and Zworski [4] are

reproduced in figure 5.11. The authors found exponents α close to the value suggested bythe fractal Weyl law (2.15). Unfortunately, it is not clear from the paper which symmetrysubspaces have been used for the discussion.

5.4.1 Configuration d/R = 6

For d/R = 6, we obtain the exponents shown in figure 5.12. For the A1 subspace, wefind a very good agreement for moderate values of C < 1.4. The relative error in thisC-interval is less than 2 %. However, in the A2 subspace, all computed exponents aretoo large by about 8 % for the same C-interval. One possible reason is that the k-rangeused for the fitting is too small.

73

5 Results

1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.550.1

0.15

0.2

0.25

0.3

0.35

0.4

strip width (2C/γ0)

Exp

on

en

t

r/a=5 r/a=6 r/a=10

Figure 5.11: Results for the two-dimensional three-disk billiard system. All calculatedexponents α are close to the expected value (from [4]). The relative error is . 10 %.


Performing the same procedure for a configuration parameter of d/R = 8, we obtain theplots shown in figure 5.13.Both plots reveal a clear tendency to obey the fractal Weyl law within a smaller

error range as the range of k values used for the fit increases. However, for reasons ofconvergence, longer spectra could not be used as non-converged resonances would beused for the counting. We note that for C < 1.4 and k ∈ [0; 3000], the error is less than7 % for the A1 resonances. The exponents obtained from A2 resonances are larger thanthe expected exponent. For C = 1.3, the relative error is about 15 %.


Finally, the system configuration given by d/R = 10 has been studied. Results are shownin figure 5.14.The counting functions for the A1 subspace once more tend to give the expected

exponent as the k-interval used for the fit increases. For k ∈ [0; 3000] and C < 1.4,

74

5.5 Discussion

the error is less than 3 %. Again, the A2 spectra yield exponents that are too large.Possibly the k-range investigated here is not large enough to exhibit the asymptoticclearly. Spectra belonging to the E, T1 and T2 group representations have not beenevaluated since the convergence of cycle expansions for these symmetry subspaces is notsufficent for broader resonances.

5.5 Discussion

We have shown that for all configurations studied in this thesis, the one-dimensionalgroup representations A1 and A2 converge well in cycle expansions. In the A1 subspace,we have found a convincing agreement of the counting functions N(k) with the suggestedpower law N(k) ∝ k1+dH for all values of the configuration parameter d/R. The errorsare comparable to those found in previous investigations of billiard systems [4]. However,we note that a very large k-range seems to be necessary for a proper investigation. Thistendency is also visible for the A2 spectra. The exponents we calculated for the A2

spectra were too large. However, using larger k-intervals, the exponents seem to approachthe correct value for large strip widths C . Possibly, if larger spectra were available, theexpected exponents could be obtained. Unfortunately, we are limited by the convergenceof the cycle expansions we use. The higher-dimensional symmetry subspaces E, T1 andT2 could not be used to put the fractal Weyl to test law since the spectra did not containenough converged resonances.

However, the need for large spectra is surprising since previous work on the four-spherebilliard has shown excellent agreement of cycle expansions and exact quantum calculationsfrom Re k ≈ 10 [8]. This agreement suggests that the classical limit ~ → 0 is alreadyvalid from low wavenumbers on. However, it is still possible that the energy range ofRe k . 6000 used in this thesis is still not enough to exhibit the asymptotic level numberstatistics.

75

5 Results

1.21

1.22

1.23

1.24

1.25

1.26

1.27

1.28

1.29

1.3

1.31

1.32

1.1 1.2 1.3 1.4 1.5

α

2C/γ0

k ∈ [0; 2000]

(a)

1.22

1.24

1.26

1.28

1.3

1.32

1.34

1.36

1.38

1.4

1.1 1.2 1.3 1.4 1.5

α

2C/γ0

k ∈ [0; 2000]

(b)

Figure 5.12: Exponents α obtained from least-squares fits of a power law to measuredcounting functions for (a) A1 resonances and (b) A2 resonances calculated for d/R = 6in order 13 of cycle expansion. The power law has been fitted to the intervalk ∈ [0; 2000]. The vertical dotted line gives the classical exponent 1 + dH = 1.2354.

76

5.5 Discussion

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

1.6

1.65

1.1 1.2 1.3 1.4 1.5

α

2C/γ0

k ∈ [0; 2000]k ∈ [0; 2500]k ∈ [0; 3000]

(a)

1.2

1.4

1.6

1.8

2

2.2

1.1 1.2 1.3 1.4 1.5

α

2C/γ0

k ∈ [0; 2000]k ∈ [0; 2500]k ∈ [0; 3000]

(b)

Figure 5.13: Exponents α obtained for d/R = 8 from least-squares fits of a power lawto measured counting functions for (a) A1 resonances and (b) A2 resonances. Thepower law has been fitted to several k-intervals. The vertical dotted line gives theclassical exponent 1 + dH = 1.2063.

77

5 Results

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.1 1.2 1.3 1.4 1.5

α

2C/γ0

k ∈ [0; 2000]k ∈ [0; 3000]k ∈ [0; 4000]k ∈ [0; 5000]k ∈ [0; 6000]

(a)

1

1.2

1.4

1.6

1.8

2

2.2

2.4

1.1 1.2 1.3 1.4 1.5

α

2C/γ0

k ∈ [0; 2000]k ∈ [0; 3000]k ∈ [0; 4000]k ∈ [0; 5000]

(b)

Figure 5.14: Exponents α obtained for d/R = 10 from least-squares fits of a power lawto measured counting functions for (a) A1 resonances and (b) A2 resonances. Resultsfor several k-intervals are shown. The vertical dotted line gives the classical exponent1 + dH = 1.1888. More than 50 000 resonances have found to be converged.

78

6 Conclusion and outlook

This thesis has aimed at a test of the fractal Weyl law for a three-dimensional scatteringsystem. The four-sphere billiard was investigated both classically and quantum mechani-cally.

In chapter 3, we have developed a fast and very precise method to gauge the repeller.We found estimates for the Hausdorff dimension dH with a relative accuracy of 10−4.Although the algorithm is based on strong assumptions, it works over a wider range ofthe configuration parameter d/R than any existing method.

In chapter 4, we have presented the methods of semiclassical quantisation. We haveapplied the method of cycle expansion to the four-sphere billiard. Furthermore, for thefirst time, the method of symmetry decomposition was demonstrated for the Gutzwiller-Voros zeta function of the system.

We have given results in the last chapter 5. We have investigated the convergence ofcycle expansions for different values of the ratio d/R. Furthermore, we have providedspectra for various symmetry subspaces. We confirmed the expectation that the conver-gence of cycle expansions worsens as d/R decreases. Also, the shadowing effect of cycleexpansion was shown to work best in the totally symmetric subspace A1. The limitationsof the method of cycle expansions have been demonstrated. Finally, we have providedtests of the fractal Weyl law for various configurations of the system. Although we havefound the counting functions N(k) to deviate from power functions, we could confirmthe fractal Weyl law for the four-sphere scattering within a small error range. For thosespectra we did not find a convincing agreement of calculated level numbers N(k) withthe prediction N(k) ∝ k1+dH for, there is hope that larger spectra would lead to theexpected exponent. We also assume that the deviations from pure power laws are dueto the fact that the energy range under consideration is too small. We note that otherauthors encounter a similar problem as well [3].

As an outlook, the physical origin of the modulations in the counting functions N(k)will have to be investigated. Moreover, it is desirable to study further three-dimensionalscattering systems.

79

6 Conclusion and outlook

80

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Acknowledgements

During my time at the university, I received a lot of support from different people. Inone way or another, these people all contributed to this thesis, be it by encouragingme, helping me or any other way of appreciated good deed. First of all, I am deeplyindebted to Prof. Dr. Jörg Main for giving me the chance to work on a very excitingtopic as well as for extraordinary supervision, stimulating discussions and a guiding hand.Also, I want to thank Prof. Dr. Günter Wunner for accepting me at the institute andProf. Dr. Günter Mahler for accepting the second supervision of this thesis. RüdigerEichler shall be thanked for being a good office mate and friend. I am grateful towardsDr. Holger Cartarius for support with any kind of problems. Morever, thanks to all theother institute members for making the institute a good place to be. Three cheers for thecoffee round! Special thanks go to Luisa for support and proof-reading. Heaps thanks tomy family – without you, it would have been far more complicated!

Declaration of authenticity

I hereby declare that I wrote this work independently and used no other sources or aidsthan those indicated.

Stuttgart, 31. März 2010 Alexander Eberspächer

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