Hilbert Space Problems

7/27/2019 Hilbert Space Problems

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Problems and SolutionsinHilbert space theory,waveletsandgeneralized functions

byWilli-Hans SteebInternational School for Scientic ComputingatUniversity of Johannesburg, South Africa


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PrefaceThe purpose of this book is to supply a collection of problems in Hilbertspace theory, wavelets and generalized functions.

Prescribed books for problems.

1) Hilbert Spaces, Wavelets, Generalized Functions and Modern QuantumMechanics

by Willi-Hans SteebKluwer Academic Publishers, 1998ISBN 0-7923-5231-9

2) Classical and Quantum Computing with C++ and Java Simulations

by Yorick Hardy and Willi-Hans SteebBirkhauser Verlag, Boston, 2002ISBN 376-436-610-0

3) Problems and Solutions in Quantum Computing and Quantum Informa-tion, second edition

by Willi-Hans Steeb and Yorick HardyWorld Scientic, Singapore, 2006ISBN 981-256-916-2

http://www.worldscibooks.com/physics/6077.html

The International School for Scientic Computing (ISSC) provides certi-cate courses for this subject. Please contact the author if you want to dothis course or other courses of the ISSC.

e-mail addresses of the author:

[email protected][email protected]

Home page of the author:

http://issc.uj.ac.za

v
http://www.worldscibooks.com/physics/6077.htmlhttp://www.worldscibooks.com/physics/6077.html


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Contents

Notation ix

1 General 1

2 Finite Dimensional Hilbert Spaces 8

3 Hilbert Space L2() 19

4 Hilbert Space 2(N) 41

5 Fourier Transform 44

6 Wavelets 48

7 Operators 52

8 Generalized Functions 56

Bibliography 68

Index 70

vii


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Notation

:= is dened asbelongs to (a set)

/ does not belong to (a set)

intersection of setsunion of setsempty set

N set of natural numbersZ set of integersQ set of rational numbersR set of real numbers

R+

set of nonnegative real numbersC set of complex numbersRn n-dimensional Euclidean space

space of column vectors with n real componentsCn n-dimensional complex linear space

space of column vectors with n complex components

H Hilbert spacei 1z real part of the complex number zz imaginary part of the complex number z

|z| modulus of complex number z|x + iy| = ( x2 + y2)1/ 2 , x , y RT S subset T of set S

S T the intersection of the sets S and T S T the union of the sets S and T f (S ) image of set S under mapping f f g composition of two mappings ( f g)(x) = f (g(x))x column vector in C nx T transpose of x (row vector)0 zero (column) vector

. normx y x y scalar product (inner product) in C nx y vector product in R 3A, B, C m n matricesdet( A) determinant of a square matrix Atr(A) trace of a square matrix Arank( A) rank of matrix AAT transpose of matrix A

ix


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A conjugate of matrix AA conjugate transpose of matrix AA conjugate transpose of matrix A

(notation used in physics)A1 inverse of square matrix A (if it exists)I n n n unit matrixI unit operator0n n n zero matrixAB matrix product of m n matrix Aand n p matrix BA B Hadamard product (entry-wise product)of m n matrices A and B[A, B ] := AB BA commutator for square matrices A and B[A, B ]+ := AB + BA anticommutator for square matrices A and BA B Kronecker product of matrices A and BA B Direct sum of matrices A and B

jk Kronecker delta with jk = 1 for j = kand jk = 0 for j = k eigenvalue

real parametert time variableH Hamilton operator

The Pauli spin matrices are used extensively in the book. They are givenby

x :=0 11 0 , y :=

0 ii 0 , z :=1 00 1

.

In some cases we will also use 1 , 2 and 3 to denote x , y and z .

x


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Chapter 1

General

Problem 1. Let Hbe a Hilbert space with scalar product , . Letu, v H.(i) Show that| u, v | u v .

(ii) Show thatu + v u + v .

Problem 2. Consider a Hilbert space Hwith scalar product , . Thescalar product implies a norm via f 2 := f, f , where f H.(i) Show thatf + g 2 + f g 2 = 2( f 2 + g 2) .

(ii) Assume that f, g = 0, where f, g H. Show thatf + g 2 = f 2 + g 2 .

Problem 3. Let f, g H. Use the Schwarz inequality

| f, g |2 f, f g, g = f 2 g 2to prove the triangle inequality

f + g f + g .1


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2 Problems and Solutions

Problem 4. Consider a complex Hilbert space Hand |1 , |2 H. Letc1 , c2 C. An antilinear operator K in this Hilbert space His characterizedbyK (c1

|1 + c2

|2 ) = c1 K

|1 + c2K

|2 .

A comb is an antilinear operator K with zero expectation value for all states

| of a certain complex Hilbert space H. This means|K | = |LC | = |L| = 0

for all states | H, where L is a linear operator and C is the complexconjugation.(i) Consider the two-dimensional Hilbert space H= C2 . Find a unitary2 2 matrix such that

|UC | = 0 .(ii) Consider the Pauli spin matrices with 0 = I 2 , 1 = x , 2 = y ,

3=

z. Find

3

=0

3

=0| C | g, | C |

where g, = diag( 1, 1, 0, 1).Problem 5. Let P be a nonzero projection operator in a Hilbert space

H. Show that P = 1.Problem 6. A family, {j }j J of vectors in the Hilbert space, H, iscalled a frame if for any f H there exist two constants A > 0 and0 < B < , such that

A f 2 j J

j |f |2 B f 2 .

Consider the Hilbert space H= R2 and the family of vectors0 =

11 , 1 =

1

1.

Show that we have a tight frame.

Problem 7. Let T : X Y be a linear map between linear spaces (vectorspaces) X , Y . The null space or kernel of the linear map T , denoted bykerT , is the subset of X dened by

kerT := {x X : T x = 0 }.


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has a unique solution u.

Problem 12. Let f, g H. Find all solutions to the equationsf, g g, f = i .

Problem 13. Let f, g H. Show thatf + g 2 + f g 2 = 2( f 2 + g 2)

where the norm is implied by the scalar product of the Hilbert space.

Problem 14. Show that

f, g =14

f + g 2 14

f g 2

or f, g = 14

f + g 2 14 f g2 + i

4f + ig 2 i4 f ig

2

depending on whether we are dealing with a real and complex Hilbert space.

Problem 15. Given a Hilbert space Hand a Hilbert subspace Gof H.The Hilbert space projection theorem states that for every f H, thereexists a unique g G such that(i) f g G

(ii ) f g = inf h G f hwhere the space

Gis dened by

G := {k H : k|u = 0 for all u G }.Show that if g is the minimizer of f h over all h G, then it is truethat f g G .Problem 16. Let {n }n Z be an orthonormal basis in a Hilbert spaceH. Then any vector f H can be written as

f =n Z

f, n n .

Now suppose that {n }n Z is also a basis for H, but it is not orthonormal.Show that if we can nd a so-called dual basis{

n

}n Z satisfying

n | m = (n m)


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then

x 2 =

j =1|cj |2 .

Problem 21. Two Cauchy sequences {xk }and {yk }are said to beequivalent if for all > 0, there is a k(epsilon ) such that for all j k( )we have d(x j , yj ) < . One writes {xk } {yk }. Obviously, is anequivalence relationship. Show that equivalent Cauchy sequences have thesame limit.

Problem 22. Consider the sequence {xk }, k = 1 , 2, . . . in R denedby kk = 1 /k 2 for all k = 1 , 2, . . . . Show that this sequence is a Cauchysequence.

Problem 23. Let

Hbe a Hilbert space and

S be a sub Hilbert space.

Show that any element u of Hcan be decomposed uniquelyu = v + w

where v is in S and w is in S .Problem 24. Let u, v1 , v2 be elements of a Hilbert space. Show that

2 u v1 2 + 2 u v2 2 = 2 u v1 + v2

22 + v1 v2 2 .

Problem 25. Let P be the set of prime numbers. We dene the setS := {( p, q ) : p, q P p q }.

Show thatd(( p1 , q 1), ( p2 , q 2)) := | p1q 1 p2q 2|

denes a metric.

Problem 26. Consider the vector space of all continuous functions de-ned on [a, b]. We dene a metric

d(f, g ) := maxa

x

b |f (x) g(x)| .

Let a = , b = , f (x) = sin( x) and g(x) = cos( x). Find d(f, g ).


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Finite Dimensional Hilbert Spaces 9

can only be +1 or 1. Calculate the matrix p

i =1

x i x T i

and nd the eigenvalues and eigenvectors of this matrix

Problem 4. A sequence {f n }(n N) of elements in a normed space E is called a Cauchy sequence if, for every > 0, there exists a number M ,such that f p f q < for p,q > M . Consider the Hilbert space R. Showthat

sn =n

j =1

1( j 1)!

, n 1is a Cauchy sequence.

Problem 5. Two Cauchy sequences {xk }and {yk }are said to be equiv-alent if for all > 0, there is a k( ) such that for all j k( ) we haved(x j , yj ) < . One writes {xk } {yk }. Obviously, is an equivalencerelationship. Show that equivalent Cauchy sequences have the same limit.Problem 6. Consider the sequence {xk }, k = 1 , 2, . . . in R denedby xk = 1 /k 2 for all k = 1 , 2, . . . . Show that this sequence is a Cauchysequence.

Problem 7. Consider the Hilbert space C2 and the vectors

|0 =ii , |1 =

1

1 .

Normalize these vectors and then calculate the probability | 0|1 |2 .Problem 8. Consider the Hilbert space Rn . Let x , y Rn . Show that

x + y 2 + x y 2 2( x 2 + y 2) .Note that

x 2 := x , y .

Problem 9. Let |0 , |1 be an orthonormal basis in the Hilbert space C2 .Let| = cos( / 2)|0 + e

i

sin(/ 2)|1where , R.


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Problem 13. Consider the Hilbert space Hof the 2 2 matrices overthe complex numbers with the scalar productA, B := tr( AB ), A, B

H.

Show that the rescaled Pauli matrices j := 1 2 j , j = 1 , 2, 3

1 =1 2

0 11 0 , 2 =

1 2

0 ii 0 , 3 =1 2

1 00 1

plus the rescaled 2 2 identity matrix0 =

1 2

1 00 1

form an orthonormal basis in the Hilbert space H.

Problem 14. Let A, B be two n n matrices over C. We introduce thescalar productA, B :=

tr( AB )tr I n

=1n

tr( AB ) .

This provides us with a Hilbert space.The Lie group SU (N ) is dened by the complex n n matrices U

SU (N ) := {U : U U = UU = I n , det( U ) = 1 }.The dimension is N 2 1. The Lie algebra su (N ) is dened by the n nmatrices X

su (N ) := {X : X = X , tr X = 0 }.(i) Let U SU (N ). Calculate U, U .(ii) Let A be an arbitrary complex nn matrix. Let U SU (N ). CalculateUA,UA .(iii) Consider the Lie algebra su (2). Provide a basis. The elements of thebasis should be orthogonal to each other with respect to the scalar productgiven above. Calculate the commutators of these matrices.

Problem 15. Let H = S x be a Hamilton operator, where

S x :=

20 1 01 0 10 1 0

and is the frequency.(i) Find exp( iHt/ )(0), where (0) = (1 , 1, 1)T / 3.


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(ii) Calculate the time evolution of

S z := 1 0 00 0 00 0 1

using the Heisenberg equation of motion. The matrices S x , S y , S z are thespin-1 matrices, where

S y :=

20 i 0i 0 i0 i 0

.

Problem 16. Consider the linear operator

A =

2 0 0

0 0 10 1 0

in the Hilbert space R3 . Find

A := supx =1

Ax

using the method of the Lagrange multiplier.

Problem 17. Consider the Hilbert space R4 . Show that the Bell basis

u 1 =1

21001

, u 2 =1

2100

1, u 3 =

1

20110

, u 4 =1

201

10forms an orthonormal basis in this Hilbert space.

Problem 18. Consider the Hilbert space R3 . Let x R3 , where x isconsidered as a column vector. Find the matrix xx T . Show that at leastone eigenvalue is equal to 0.

Problem 19. (i) Consider the Hilbert space C4 . Show that the matrices

1 =12

(I 2 I 2 + x x ), 2 =12

(I 2 I 2 x x )are projection matrices in C4 .


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the Hilbert space Rn . Show that u j v k ( j = 1 , 2, . . . , m ), (k = 1 , 2, . . . , n )is an orthonormal basis in Rm + n .

Problem 26. Show that the 2

2 matrices

A =1 2

1 00 1 , B =

1 2

0 11 0 ,

C =1 2

0 ii 0 , D =1 2

1 00 1

form an orthonormal basis in the Hilbert space M 2(C).

Problem 27. Show that the 2 2 matricesA = 1 00 0 , B =

1 10 0 , C =

1 11 0 , D =

1 11 1

form a basis in the Hilbert space M 2(R). Apply the Gram-Schmidt tech-nique to obtain an orthonormal basis.

Problem 28. Consider the 3 3 matrices over the real numbers

A =2 0 21 0 00 0 1

.

(i) The matrix A can be considered as an element of the Hilbert space of the 3 3 matrices over the real nunbers with the scalar product

B, C := tr( BC T ) .

Find the norm of A with respect to this Hilbert space.(ii) On the other hand the matrix A can be considered as a linear operatorin the Hilbert space R3 . Find the norm

A := supx =1

Ax , x R3 .

(iii) Find the eigenvalues of A and AT A. Compare the result with (i) and(ii).

Problem 29. Consider the Hilbert space C2 . The Pauli spin matricesx , y , z act as linear operators in this Hilbert space. Let

H = z


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and D(A) = 2(N ). Show that A is self-adjoint if the n n matrix M ishermitian. Show that A is unitary if M is unitary.Problem 32. Consider the Hilbert space Cn . Let u

j, j = 1 , 2, . . . , n , and

v j , j = 1 , 2, . . . , n be orthonormal bases in Cn , where u j , v j are consideredas column vectors. Show that

U =n

j =1

u j v j

is a unitary n n matrix.Problem 33. Consider the Hilbert space R2 . Given the vectors

u 1 =01 , u 2 =

3/ 21/ 2

, u 3 = 3/ 21/ 2.

The three vectors u 1 , u 2 , u 3 are at 120 degrees of each other and arenormalized, i.e. u j = 1 for j = 1 , 2, 3. Every given two-dimensionalvector v can be written as

v = c1u 1 + c2u 2 + c3u 3 , c1 , c2 , c3 R

in many different ways. Given the vector v minimize

12

(c21 + c22 + c

23)

subject to the two constraints

v c1u 1 c2u 2 c3u 3 = 0 .

Problem 34. Let A, H be n n hermitian matrices, where H plays therole of the Hamilton operator. The Heisenberg equations of motion is givenby

dA(t)dt

=i [H, A (t)] .

with A = A(t = 0) = A(0). Let E j ( j = 1 , 2, . . . , n 2) be an orthonormalbasis in the Hilbert space Hof the n n matrices with scalar product

X, Y := tr( XY ), X, Y H.Now A(t) can be expanded using this orthonormal basis as

A(t) =n 2

j =1cj (t)E j


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Chapter 3

Hilbert Space L 2()

Problem 1. A basis in the Hilbert space L2[0, 1] is given by

B := e2ixn : n Z .

Let

f (x) = 2x 0 x < 1/ 22(1 x) 1/ 2 x < 1Is f L2[0, 1]? Find the rst two expansion coefficients of the Fourierexpansion of f with respect to the basis given above.

Problem 2. (i) Consider the Hilbert space L2[1, 1]. Consider the se-quencef n (x) =

1 if 1 x 1/nnx if 1/n x 1/n+1 if 1/n x 1where n = 1 , 2, . . . . Show that {f n (x) }is a sequence in L2[1, 1] that is aCauchy sequence in the norm of L2[1, 1].(ii) Show that f n (x) converges in the norm of L2[1, 1] to

sgn(x) = 1 if 1 x < 0+1 if 0 < x 1.

(iii) Use this sequence to show that the space C [1, 1] is a subspace of L2[1, 1] that is not closed.19


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Problem 3. Let f L2(R). Give the denition of the Fourier transform.Let us call the transformed function f . Is f L2(R)? What is preservedunder the Fourier transform?

Problem 4. Consider the Hilbert space L2[a, b], where a, b R and b > a .Find the condition on a and b such that

cos(x), sin(x) = 0

where , denotes the scalar product in L2[a, b].Hint. Since b > a , we can write b = x + , where > 0.

Problem 5. Consider the Hilbert space L2[0, 1]. The Legendre polyno-mials are dened as

P 0(x) = 1 , P n (x) =1

2n n!dn

dxn(x2 1).

Show that the rst rst four elements are given by

P 0(x) = 1 , P 1(x) = x, P 2(x) =12

(3x2 1), P 3(x) =12

(5x3 3x) .Normalize the four elements. Show that the four elements are pairwiseorthonormal.

Problem 6. Let R be a bounded region in n-dimensional space. Considerthe eigenvalue problem

u = u, u (q R ) = 0where R denotes the boundary of R.(i) Show that all eigenvalues are real and positive(ii) Show that the eigenfunctions which belong to different eigenvalues areorthogonal.

Problem 7. Consider the inner product space

C [a, b] = {f (x) : f is continuous on x [a, b]}with the inner product

f, g := b

af (x)g (x)dx .

This implies a norm

f, f = b

af (x)f (x)dx = f 2 .


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Hilbert Space L2() 25

(iii) Show that {f n }(n = 1 , 2, . . . ) is an orthonormal system. Is it a basisin the Hilbert space L2[0, )?

Problem 22. Consider the function f L2[0, 1]

f (x) = x for 0 x 1/ 21 x for 1/ 2 x 1A basis in the Hilbert space is given by

B:= 1, 2 cos(nx ) : n = 1 , 2, . . . .Find the Fourier expansion of f with respect to this basis. From thisexpansion show that

2

8=

k=0

1(2k + 1) 2

.

Problem 23. A particle is enclosed in a rectangular box with impene-trable walls, inside which it can move freely. The Hilbert space is

L2([0, a ][0, b][0, c])where a, b, c > 0. Find the eigenfunctions and the eigenvalues. What canbe said about the degeneracy, if any, of the eigenfunctions?

Problem 24. Consider the Hilbert space L2[0, 1]. Find a non-trivialfunction

f (x) = ax 3 + bx2 + cx + d .

such thatf (x), x = 0 , f (x), x2 = 0 , f (x), x3 = 0

where , denotes the scalar product

Problem 25. Consider the Hilbert space L2[0, 1]. Find a non-trivialfunction f such that

f (x), x = 0 , f (x), x2 = 0 , f (x), x3 = 0

where , denotes the scalar product

Problem 26. Consider the Hilbert space L2[0, 1] and the polynomials

1, x, x2 , x3 , x4 .


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Problem 38. The Legrendre polynomials are dened on the interval[1, 1] and dened by the recurrence formula

L j (x) =2 j + 1 j + 1 xL j (x)

j j + 1 L j 1(x) j = 1 , 2, . . .

and L0(y) = 1 , L1(x) = x. They are elements of the Hilbert spaceL2([1, 1]). Calculate the scalar product

L j (x), Lk (x)

for j, k = 0 , 1, . . . . Discuss.

Problem 39. Let f n : [1, 1] [1, 1] be dened by

f n (x) =1 for 1 x 0 1

nx for 0

x

1/n

0 for 1/n x 1Show that f n L2[1, 1]. Show that f n is a Cauchy sequence.Problem 40. Consider the Hilbert space L2([1, 1]). The Chebyshev polynomials are dened by

T n (x) := cos( n cos1 x), n = 0 , 1, 2, . . . .

They are elements of the Hilbert space L2([1, 1]). We haveT 0(x) = 1 , T 1(x) = x, T 2(x) = 2 x2 1, T 3(x) = 4 x3 3x .

Calculate the scalar products

T 0 , T 1 , T 1 , T 2 , T 2 , T 3 .

Calculate the integrals

1

1T m (x)T n (x) 1 x2

dx

for (m, n ) = (0 , 1), (m, n ) = (1 , 2), (m, n ) = (2 , 3).

Problem 41. (i) Consider the Hilbert space L2[0, 1] with the scalarproduct , . Let f : [0, 1] [0, 1]

f (x) := 2x if x [0, 1/ 2)2(1 x) if x [1/ 2, 1]


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and U n (e) = I , where e is the identity element in SU (1, 1) (2 2 unitmatrix).Show that

1(1 |z|2)2 dx dy

is invariant z zg.Problem 44. Consider the problem of a particle in a one-dimensionalbox. The underlying Hilbert space is L2(a, a ). Solve the Schr odingerequation

i t

= H

as follows: The formal solution is given by

(t) = exp( iHt/ )(0) .Expand (0) with respect to the eigenfunctions of the operator H . Theeigenfunctions form a basis of the Hilbert space. Then apply exp( iHt/ ).Calculate the probability

P = | , (t) |2where

(q ) =1 a sin

q a

and(q, 0) =

1 a sin

q a

.

Problem 45. Let f L2(Rn ). Consider the following operators

T y f (x ) = f (x y ), translation operatorM k f (x ) = ei x k f (x ), modulation operatorD s f (x ) = |s|n/ 2f (s1x ), s R \ {0} dilation operator

where x k = k1x1 + + xn kn .(i) Find T y f , M k , D s f , where denotes the norm in L2(Rn ).(ii) Find the adjoint operators of these three operators.

Problem 46. Consider the vector space

H 1(a, b) := {f (x) L2(a, b) : f (x) L2(a, b) }


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with the norm g H 1(a, b))

g 1 :=

g 20 + g/x 20 .

Consider the Hilbert space L2(, ) and f (x) = sin( x). Find the normf 1 .

Problem 47. Let f H 1(a, b). Then for a x < y b we havef (y) = f (x) +

y

xf (s)ds .

(i) Show that f C [a, b].(ii) Show that

|f (y) f (x)| f 1 |y x| .Problem 48. Consider the Hilbert space L2[0, ). The Laguerre poly-nomials are dened by

Ln (x) = exdn

dxn(xn ex ), n = 0 , 1, 2, . . . .

The rst ve Laguerre polynomials are given by

L0(x) = 1L1(x) = 1 xL2(x) = 2 4x + x2L3(x) = 6 18x + 9 x2 x3L4(x) = 24

96x + 72 x2

16x3 + x4 .

Show that the function

n (x) =1n!

ex/ 2Ln (x)

form an orthonormal system in the Hilbert space L2[0, ).Problem 49. Consider the Hilbert space L2[, ]. A basis in thisHilbert space is given by

B=1

2 eikx : k Z .

Find the Fourier expansion of

f (x) = 1 .


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(i) Calculate the scalar product

B mi (x), Bnj (x) .

(ii) Find the coefficients cij .

Problem 52. Consider Fourier series and analytic (harmonic) functionson the disc

D := {z C : |z| 1}.A Fourier series can be viewed as the boundary values of a Laurent series

n = cn zn .

Suppose we are given a function f on T. Find the harmonic extension u of f into D. This means

u = 0 and u = f on D = T

where := 2 /x 2 + 2 /y 2 .

Problem 53. Consider the compact abelian Lie group U (1)

U (1) = {e2i : 0 < 1}.The Hilbert space L2(U (1)) is the space L2([0, 1]) consisting of all measure-able funcrions f () with period 1 such that

1

0 |f ()|2

d < .Now the set of functions

{e2im : m Z }form an orthonormal basis for the Hilbert space L2([0, 1]). Thus everyf L2([0, 1]) can be expressed uniquely as

f () =+

m = cm e2im , cm =

1

0f ()e2im d .

Calculate

1

0 |f ()|2 .


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Problem 54. The Hilbert space L2(R) is the vector space of measur-able functions dened almost everywhere on R such that |f |2 is integrable.H 1(R) is the vector space of functions with rst derivatives in L2(R). Givetwo examples of such a function.

Problem 55. Consider the Hilbert space L2[, ]. The set of functions1

2 einx

n Z

is an orthonormal basis for L2[, ]. LetK (x, t ) =

1 2 e

itx .

For t xed nd the Fourier expansion of this function.

Problem 56. Consider the vector space C ([0, 1]) of continouos functions.We dene the triangle function

(x) := 2x 0 x 1/ 22 2x 1/ 2 < x 1.

Let 0(x) := x andn (x) := (2 j x k)

where j = 0 , 1, 2, . . . , n = 2 j + k and 0 k < 2j . The functions{1, 0 , 1 , . . . }

are the Schauder basis for the vector space C ([0, 1]). Let f C ([0, 1]).Then

f (x) = a + bx +

n =1cn n (x) .

(i) Find the Schauder coefficients a, b, cn .(ii) Consider g : [0, 1] [0, 1]

g(x) = 4 x(1 x) .Find the Schauder coefficients for this function.

Problem 57. Let s be a nonnegative integer. Let x R and hn (n =0, 1, 2, . . . be

hn (x) = (1)n2n/ 2 n! 4 exp( x2 / 2) dn ex

2

dxn.


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Problem 61. Consider the Hilbert space L2(R2) with the basis

mn (x1 , x2) = NH m (x1)H n (x2)e(x21 + x

22 ) / 2

where m, n = 0 , 1, . . . and N is the normalization factor. Consider thetwo-dimensional potential

V (x1 , x2) =a4

(x41 + x42) + cx1x2 .

(i) Find all linear transformation T : R2 R2 such thatV (T x ) = V (x ) .

(ii) Show that these 2 2 matrices form a group. Is the group abelian.(iii) Find the conjugacy classes and the irreducible representations.(iv) Consider the Hilbert space L2(R2) with the orthogonal basis

mn (x1 , x2) = H m (x1)ex21 / 2H n (x2)ex

22 / 2

where m, n = 0 , 1, 2, . . . . Find the invariant subspaces from the projectionoperators of the irreducible representations.

Problem 62. Consider the Hilbert space L2[, ]. A basis in thisHilbert space is given by

B=1

2 eikx : k Z

Find the Fourier expansion of

f (x) = 1 .

Problem 63. Consider the Hilbert space L2[0, 1]. Let P n be the n + 1-dimensional real linear space of all polynomial of maximal degree n in thevariable x, i.e.

P n = span {1,x ,x 2 , . . . , x n }.The linear space P n can be spanned by various systems of basis functions.An important basis is formed by the Bernstein polynomials {B n0 (x), B n1 (x), . . . , B nn (x)}of degree n with

B ni (x) := xi (1 x)n i , i = 0 , 1, . . . , n .

The Bernstein polynomials have a unique dual basis {D(0x), D n1 (x), . . . , D nn (x) }which consists of the n + 1 dual basis functions

D ni (x) =n

j =0cij B nj (x) .


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The dual basis functions satisfy

D ni (x), Bnj (x) = ij .

(i) Find the scalar productB mi (x), B

nj (x) .

(ii) Find the coefficients cij .

Problem 64. Let V be a metric vector space. A reproducing kernel Hilbert space on V is a Hilbert space Hof functions on V such that for eachx V , the point evaluation functional

x (f ) := f (x), f His continouos. A reproducing kernel Hilbert space Hpossesses a uniquereproducing kernel K which is a function on V

V characterized by the

properties that for all f H and x V , K (x, ) H andf (x) = f, K (x, ) H

where , H denotes the inner product on H. The reproducing kernel K uniquely determines the reproducing kernel Hilbert space H. The repro-ducing kernel Hilbert space of a reproducing kernel K is denoted by HK .The Paley-Wiener space is dened byS := {f C (Rd ) L2(Rd ) : supp f [, ]d }

is a reproducing kernel Hilbert space. The Fourier transform of f L1(Rd )is given by

f (k ) := 1( 2)2d R 2 d f (x )e

i x k dx , k Rd

where x k = x1k1 + + xd kd is the inner product in Rd . The norm on thevector space S inherits from that in L2(Rd ). Show that the reproducingkernel for the Paley-Wiener space S is the sinc function

sinc(x , y ) :=d

j =1

sin((x j yj ))(x j yj )

, x , y Rd .

Problem 65. Can one construct an orthonormal basis in the Hilbertspace L2(R) starting from ( > 0)

e|x |/ , xe|x |/ , x2e|x |/ , x3e|x |/ , . . . .


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Chapter 4

Hilbert Space 2(N)

Problem 1. Consider the Hilbert space 2(N). Let x = ( x1 , x2 , . . . )T bean element of 2(N). We dene the linear operator A in 2(N) as

Ax = ( x2 , x3 , . . . )T

i.e. x1 is omitted and the n+1st coordinate replaces the nth for n = 1 , 2, . . . .Then for the domain we have D(A) = 2(N). Find A y and the domain of A , where y = ( y1 , y, . . . ). Is A unitary?Problem 2. Consider the Hilbert space 2(N) and x = ( x1 , x2 , . . . )T . Thelinear bounded operator A is dened by

A(x1 , x2 , x3 , . . . , x 2n , x2n +1 , . . . )T = ( x2 , x4 , x1 , x6 , x3 , x8 , x5 , . . . , x 2n +2 , x2n 1 , . . . )T .

Show that the operator A is unitary. Show that the point spectrum of A isempty and the continuous spectrum is the entire unit circle in the -plane.

Problem 3. Consider the Hilbert space 2(N). Suppose that S and T are the right and left shift linear operators on this sequence space, denedby

S (x1 , x2 , . . . ) = (0 , x1 , x2 , . . . ), T (x1 , x2 , x3 , . . . ) = ( x2 , x3 , x4 , . . . ) .

Show that T = S .

41


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Hilbert Space 2(N) 43

does not exist. If the linear operator H is self-adjoint, the spectrum is asubset of the real axis. The Lebesgue decomposition theorem states that

= pp ac sing

where pp is the countable union of points (the pure point spectrum), acis absolutely continuous with respect to Lebesgue measure and sing issingular with respect to Lebesgue measure, i.e. it is supported on a set of measure zero. Consider the Hilbert space 2(Z) and the linear operator

H = I 2 I 2 3 1 3 I 2 I 2 where 1 is at position 0. Find the spectrum of this linear operator.


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Chapter 5

Fourier Transform

Problem 1. Consider the Hilbert space L2(R). Find the Fourier trans-form of the function

f (x) =1 if 1 x 0ex if x 00 otherwise

Problem 2. (i) Find the Fourier transform for

f (x) =2

exp( |x|), > 0.Discuss large and small.(ii) Calculate

f (x)dx .Problem 3. Find the Fourier transform of the hat function

f (t) = 1 |t| for 1 < t < 10 otherwise

Problem 4. Let f L2(R) and f L1(R. Assume that f (x) = f (x).Can we conclude that f (k) = f (k)?44


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Fourier Transform 45

Problem 5. Consider the Hilbert space L2(R). Find the Fourier trans-form of

f (x) = ea |x |, a > 0 .

Problem 6. Consider the Hilbert space L2(R). Let a > 0. Dene

f a (x) =1

2a |x| < a0 |x| > aCalculate

R f a (x)dxand the Fourier transform of f a . Discuss the result in dependence of a.

Problem 7. Consider the Hilbert space L2(R). Let

() =1 if 1/ 2

|

| 1

0 otherwise

and() = e ||, > 0 .

(i) Calculate the inverse Fourier transform of () and (), i.e.

(t) =1

2 R eit ()(t) =

12 R eit ()

(ii) Calculate the scalar product (t)|(t) by utilizing the identity2 (t)|(t) =

()|

() .

Problem 8. Consider the Hilbert space L2(R) and the function f L2(R)

f (x) = 1 if |x| < 10 if |x| 1Calculate f f and verify the convolution theorem

f f = f f .

Problem 9. Let

f () = (1 2) for || 10 for || > 1


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Find f (t).

Problem 10. Let a > 0. Find the Fourier transform of the functionf

a: R

R

f a (x) =x/a 2 + 1 /a for a x 0x/a 2 + 1 /a for 0 x a0 otherwise

Problem 11. Let a > 0. Find the Fourier transform of

f a (t) =1 a e

a |t | .

Discuss the cases a large and a small. Is f a L2(R).

Problem 12. Show that the Fourier transform of the rectangular windowof size N wn =

1 for 0 n N 10 otherwiseis

W (ei ) =sin(N/ 2)sin(/ 2)

ei (N 1) / 2 .

Problem 13. Consider the Hilbert space L2(R). Let T > 0. Considerthe function in L2(R)

f (t) = A cos(t) for T < t < T 0 otherwisewhere A is a positive constant. Calculate the Fourier transform.

Problem 14. Let > 0. Show that the Fourier transform of the Gaussianfunction

g (x) =1

2 exp x2

22

is again a Gaussian function

g (k) = e2 k 2 / 2 .

We have

g (x)dx = 1. Is

g k (k)dk = 1 ?


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Chapter 6

Wavelets

Problem 1. Consider the Hilbert space L2[0, 1] and the function f (x) =x2 in this Hilbert space. Project the function f onto the subspace of L2[0, 1]spanned by the functions (x), (x), (2x), (2x 1), where

(x) := 1 for 0 x < 10 otherwise

(x) :=1 for 0 x < 1/ 21 for 1/ 2 x < 10 otherwise

.

This is related to the Haar wavelet expansion of f . The function is calledthe father wavelet and is called the mother wavelet.

Problem 2. Consider the function H L2(R)

H (x) =1 0 x 1/ 21 1/ 2 x 10 otherwise

LetH mn (x) := 2 m/ 2H (2m x n)

where m, n Z. Draw a picture of H 11 , H 21 , H 12 , H 22 . Show that

H mn (x), H kl (x) = mk nl , k, l Z

where . denotes the scalar product in L2(R) Expand the function

f (x) = exp( |x|)48


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Wavelets 49

with respect to H mn . The functions H mn form an orthonormal basis inL2(R).

Problem 3. Consider the Hilbert space L2[0, 1] and the Haar scaling function (father wavelet)

(x) = 1 if 0x < 10 otherwiseLet n be a positive integer. We dene

gk (x) := n(nx k), k = 0 , 1, . . . , n 1 .(i) Show that the set of functions {g0 , g1 , . . . , gn 1 }is an orthonormal setin the Hilbert space L2[0, 1].(ii) Let f be a continuous function on the unit interval [0 , 1]. Thus f L2[0, 1]. Form the projection f n on the subspace S n of the Hilbert space

L2[0, 1] spanned by {g0 , g1 , . . . gn 1 }, i.e.f n =

n 1

k =0

f, g k gk .

Show that f n (x) f (x) pointwise in x as n .Problem 4. The continuous wavelet transform

W f (a, b) =1a

+

f (t)

t ba

dt, (a, b R, a > 0)

decomposes the function f L2(R) hierarchically in terms of elementarycomponents (( tb)/a ). They are obtained from a single analyzing wavelet applying dilations and translations . Here denotes the complex conju-gate of and a is the scale and b the shift parameter. The function hasto be chosen so that it is well localized both in physical and Fourier space.The signal f (t) can be uniquely recovered by the inverse wavelet transform

f (t) =1

C +

+

0Wf (a, b)

t ba

daa

db

if (t) (respectively its Fourier transform () satises the admissibility condition

C = +

0|()|2

d < .

Consider the function(t) = tet

2 / 2 .


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Does satises the admissibility condition?

Problem 5. Consider the function H L2(R)

H (x) =1 0 x < 1/ 21 1/ 2 x 10 otherwise

LetH mn (x) := 2 m/ 2H (2m x n)

where m, n Z. Draw a picture of H 11 , H 21 , H 12 , H 22 . Show that

H mn (x), H kl (x) = mk nl , k, l Z

where . denotes the scalar product in L2(R). Expand the function

f (x) = exp( |x|)with respect to H mn . The functions H mn form an orthonormal basis inL2(R).

Problem 6. Consider the Hilbert space L2(R). Let L2(R) andassume that satises

R (t)(t k)dt = 0,ki.e. the integral equals 1 for k = 0 and vanishes for k = 1 , 2, . . . . Show thatfor any xed integer j the functions

jk (t) := 2 j/ 2(2j t k), k = 0 , 1, 2, . . .form an orthonormal set.

Problem 7. Consider the function : R R(x) := 1 for x [0, 1]0 otherwise

Find (x) := (2x) (2x 1). Calculate

(x)dx .Problem 8. Consider the Littlewood-Paley orthonormal basis of wavelets.The mother wavelet of this set is

L(x) :=1

x(sin(2 x ) sin(x )) .


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Operators 53

Solve the Schr odinger equation, where the initial function (t = 0) = (x)is given by

(x) = x/a2 + 1 /a for a x 0

x/a 2 + 1 /a for 0

x

a

Normalize . Calculate the probability to nd the particle in the state

(x) =1 a sin

xa

after time t. A basis in the Hilbert space L2[a, a ] is given by1 a sin

nxa

,1 a cos

(n 1/ 2)xa

n = 1 , 2, . . . .

Problem 5. Show that in one dimensional problems the energy spectrumof the bound state is always non-degenerate. Hint. Suppose that the oppsite

is true. Let u1 and u2 be two linearly independent eigenfunctions with thesame energy eigenvalues E , i.e.

d2u1dx2

+2m 2 (E V )u1 = 0 ,

d2u2dx2

+2m 2 (E V )u2 = 0 .

Problem 6. A particle is enclosed in a rectangular box with imprenetra-ble walls, inside which it can move freely. The Hilbert space is L2([0, a ] [0, b] [0, c]). Find the eigenfunctions and eigenvalues. What can be saidabout the degeneracy, if any, of the eigenfunctions.Problem 7. Conside the Hilbert space L2[0, 1] and the linear operator

T : L2[0, 1] L2[0, 1] dened by(T f )(x) := xf (x) .

Show that T is self-adjoint and positive denite. Find its positive squareroot.

Problem 8. Consider the Hilbert space 2(N) and the linear operator T dened by

T : (x1 , x2 , x3 , . . . ) (0, 0, x3 , x4 , . . . ) .Is T bounded? Is T self-adjoint? If so is T positive?

Problem 9. In classical mechanics we have

L = r p , T = r F


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where T is the torque, F = V (V potential depending only on r ) anddLdt

= T .

In quantum mechanics with p i , r r and wave function wehaveL = i R 3 d3x (r )

andT = R 3 d3x (r V )

since F = V . and obey the Schr odinger equation

i t

= 2

2m2 + V

i t = 2

2m2 + V .

Show thatdLdt

= T .

Problem 10. Let H be a bounded self-adjoint Hamilton operator withnormalized eigenfunctions j ( j I ) which form an orthonormal basis inthe underlying Hilbert space. We can write

(t) =j I

cj eiE j t/ j

where E j are the eigenvalues of H . Find P (t) = (t = 0) |(t) .Problem 11. Consider the Hamilton operator

H = 2

2md2

dx2+ D (1 ex )2 + eEx cos(t)

where > 0. Find the quantum Liouville equation for this Hamiltonoperator.

Problem 12. Consider the Schrodinger equation

i t

= 12m + V (x)


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Operators 55

Find the coupled system of partial differential equations for

:= , v :=

.

Problem 13. Consider the Hilbert space L2(R). Let f L2(R) and R. We dene the operator U () as

U ()f (x) := ei/ 2f (xe i ) .

Is the operator U () unitary?

Problem 14. Consider the Hilbert space L2(R). Let k Z. For k = 0we dene s0 = 0, for k 1 we dene

sk := 1 +12 +

13 + +

1k

and for k < 0 we dene sk = sk . Let > 0. Dene the indicatorfunctions W k asW k (x) :=

1 for sk < x/ sk+10 otherwiseLet u L2(R). Dene the linear operator O as

(Ou )(x) := g(x)u(x)

whereg(x) =

x

+k Z

sk + sk+1

2W k (x) .

(i) Show that O is a bounded self-adjoint operator for any > 0.(ii) Show that the norm of O

O = supu =1

Ou

is given by 1/ 2.


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an element of D(a)?Problem 5. Given a function (signal) f (t ) = f (t1 , t2 , . . . , t n ) L2(Rn )

of n real variables t = ( t1 , t2 , . . . , t n ). We dene the symplectic tomogram associated with the square integrable function f

w(X , , ) =n

k=1

12| k | R n dt1dt2 dtn f (t )exp

n

j =1

i j2 j

t2j iX j j

t j

2

where ( j = 0 for j = 1 , 2, . . . , n )

X = ( X 1 , X 2 , . . . , X n ), = ( 1 , 2 , . . . , n ), = ( 1 , 2 , . . . , n ) .

(i) Prove the equality

R n w(X , , )dX = R n |f (t )|

2

dt (1)

for the special case n = 1. The tomogram is the probabilty distributionfunction of the random variable X . This probability distribution functiondepends on 2 n extra real parameters and .(ii) The map of the function f (t ) onto the tomogram w(X , , ) is invert-ible. The square integrable function f (t ) can be associated to the densitymatrix

f (t , t ) = f (t )f (t ) .

This density matrix can be mapped onto the Ville-Wigner function

W (q , p ) =

R n

f q +u

2, q

u

2eip u du .

Show that this map is invertible.(iii) How is the tomogram w(X , , ) related to the Ville-Wigner function?(iv) Show that the Ville-Wigner function can be reconstructed from thefunction w(X , , ).(v) Show that the density matrix f (t )f (t ) can be found from w(X , , ).

Problem 6. Starting with the set of polynomials {1,x ,x 2 , . . . , x n , . . . }use the Gram-Schmidt procedure the scalar product (inner product)

f, g =

1

1(f (x)g(x) + f (x)g (x))dx

to nd the rst ve orthogonal polynomials, where f denotes derivative.


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Generalized Functions 59

Problem 7. Describe the one-dimensional scattering of a particle incidenton a Dirac delta function, i.e.

U (q ) = U 0 (q )

where u0 > 0. Find the transmission and reection coefficient.

Problem 8. (i) Give the denition of the current density, transmissioncoefficient, and reection coefficient.(ii) Calculate the transmission and the reection coefficients of a particlehaving total energy E , at the potential barrier given by

V (x) = a (x), a > 0


12

k = e

ikx=

k= (x 2k )

in the sense of generalized functionsHint. Expand the 2 periodic function

f (x) =12

x2

into a Fourier series.

Problem 10. (i) Give the denition of a generalized function.(ii) Calculate the rst and second derivative in the sense of generalizedfunction of

f (x) =0 x < 0

4x(1 x) 0 x 10 x > 1(iii) Calculate the Fourier transform of f (x) = 1 in the sense of generalizedfunctions.

Problem 11. Consider the generalized function

f (x) = |cos(x)| .Find the derivative in the sense of generalized functions.

Problem 12. Consider the generalized function

f (x) := cos(x) for x [0, 2)0 otherwise


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Find the rst and second derivative of f in the sense of generalized func-tions.

Problem 13. Find the derivative of f : R

R

f (x) = |x|in the sense of generalized functions.

Problem 14. Find the rst three derivatives of the function f : R Rdened byf (x) = e|x |

in the sense of generalized functions.

Problem 15. The Sobolev space of order m, denoted by H m (), is denedto be the space consisting of those functions in the Hilbert space L2() that,

together with all their weak partial derivatives up to and including thoseof order m, belong to the Hilbert space L2(), i.e.

H m () := {u : D u L2() for all such that | | m }.We consider real-valued functions only, and make H m () an inner productspace by introducing the Sobolev inner product , H m dened by

u, v H m := | |m (D u)(D v)dx for u, v H m () .

This inner product generates the Sobolev norm H m dened by

u2H m = u|u H m = | |m (D

u)

2dx .

Thus H 0() = L2(). We can write

u, v =

| |mD , D v L 2 ()

In other words the Sobolev inner product u, v H m () is equal to the sumof the L2() inner products of D u and D v over all such that | | m.(i) Consider the domain = (0 , 2) and the function

u(x) = x2 0 < x 12x2

2x + 1 1 < x < 2 .

Obviously u L2(). Find the Sobolev space to which u belongs.


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(ii) Find the norm of u.

Problem 16. Let c > 0. Consider the Schrodinger equation

22m d2dx2 + c (n ) (x) = Ewhere (n ) (n = 0 , 1, 2, . . .) denotes the n-th derivative of the delta function.Derive the joining conditions on the wave function .

Problem 17. The Morlet wavelet consists of a plane wave modulated bya Gaussian, i.e.

() =1

1/ 4ei e

2 / 2

where is the dimensionless frequency. Show that if = 6 the admissibilitycondition is satised.

Problem 18. Let f 0(x) = exp( x2 / 2) .We dene the mother wavelets f n as

f n (x) = d

dxf n 1(x), n = 1 , 2, . . .

Show that the family of f n s obey the Hermite recursion relation

f n (x) = xf n 1(x) (n 1)f n 2(x), n = 2 , 3, . . . .

Problem 19. Show that the 2-dimensional complex -function can bewritten as ( C)

(2)

(z) =1

2 C d2 exp( z z ) =

12 C d

2 exp( i( z + z )) .


(x x ) =1

1 + 2

k=1

cos(kx) cos(kx ) .

Problem 21. Let a > 0. Show that

m = exp( i2m (x + q )/a ) a

k = (x + q ka ) .


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with the boundary conditions u(1, ) = 0. Here is the coupling con-stant and determines the penetrability of the potential barrier. Find theeigenfunctions and the eigenvalues.

Problem 31. Show that in the sense of generalized functions

(x) =12

lim0

1e|x |/

(x) =1

lim

sin2( x )( x )2

(x) =14

lim0

11 + |x| e|x |/ .

Problem 32. Give two interpretations of the series of derivatives of functions

f (k) = 2 n =0

cn (1)n (n ) (k) . (1)


H (x a) = du

d 2

exp( iu ( x)) .


z

1z

= (z)

where

z 12

x

+ i

y.

Problem 35. Let a > 0. Show that

(x2 a2) =1

2a( (x a) + (x + a)) .

Problem 36. (i) Show that the Fourier transform in the sense of gener-alized function of the Dirac comb

n Z (x n)


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is again a Dirac comb.(ii) Find the Fourier transform in the sense of generalized functions of

1 + 2 (x) .

Problem 37. (i) Consider the nonlinear differential equation

3ududx

= 2dudx

d2udx2

+ ud3udx3

.

Show that u(x) = e|x | is a solution in the sense of generalized function.(ii) Consider the nonlinear partial differential equation

ut

3ux 2t

+ 3 uux

= 2ux

2ux 2

+ u 3ux 3

.

Show that u(x, t ) = c exp(|x ct|) (peakon ) is a solution in the sense of generalized functions.

Problem 38. Let f be a differentiable function with a simple zero at x =a such that f (x = a) = 0 and df (x = a)/dx = 0. Let g be a differentiablefunction with a simple zero at x = b = a such that g(x = b) = 0 anddg(x = b)/dx = 0. Show that

(f (x)g(x)) =1

|f (a)g(a)| (x a) +

1

|f (b)g (b)| (x b)

where denotes differentiation.

Problem 39. Consider the non-relativistic hydrogen atom, where a0 isthe Bohr radius and a = a0/Z . The Schr odinger-Coulomb Green functionG(r 1 , r 2 ; E ) corresponding to the energy variable E is the solution of thepartial differential equation

2

2m21

2

amr 1 E G(r 1 , r 2 ; E ) = (r 1 r 2)with the appropiate boundary conditions. Show that expanding G in termsof spherical harmonics Y m

G(r 1 , r 2 ; E ) =

=0 m = g (r 1 , r 2 ; E )Y m (1 , 1)Y m (2 , 2)

we nd for the radial part g of the Schr odinger-Coulomb Green function

1r 21

ddr 1

r 21 ddr 1 ( + 1)

r 21+ 2

ar 1 1

2a2g (r 1 , r 2 ; ) = 2m 2 (r

1 r 2)r 1r 2


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Fuhrmann, P. A.A Polynomial Approach to Linear Algebra Springer-Verlag, New York (1996)

Golub, G. H. and Van Loan C. F.Matrix Computations

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