## Mathematical Methods in Physics and Engineering, Winter quarter 2001:

Office hours: M: 2-3, WF:11-12 and by appointment

Office: APM 5256, tel. 534-2734

Course material: It is planned to cover most of Chapters 3,4,5,6,7,8 from the book `Mathematics of classical and quantum physics' by F.W. Byron and R.W. Fuller. Here are some possible supplementary books: L. Debnath and P. Mikusinski, Intro to Hilbert spaces; E. Zeidler, Applied functional analysis; J.T. Oden and L.F. Demkowicz, Applied functional analysis.

Computation of grade: Final: 60% Homeworks: 40%

Homeworks to be turned in on: 1/21, 2/4, 2/18, 3/4

Date of final: 3/14, 3-6pm

Homework assignments Homeworks need to be turned in on or before the listed day. Selected problems of the assignment will be graded.

Disclaimer: I will try to get the homework assignment on the net in time. Due to time and other limitations, this may not always be possible. The fact that there is no assignment posted for a particular date does therefore NOT necessarily mean that no homework is due.

for 1/21: p. 136: 3, 11(you are allowed to use properties of determinants stated in the book), 13, 17, p. 206: 7, 9, and

(a) Let T: V -> V be a linear operator whose 3rd power is equal to the identity map I, i.e. T^3x= T(T(Tx))=x for all x in V. Show: (i) the only possible eigenvalues of T are 1, t, t^2, where t=exp(2 \pi i/3). (ii) T is diagonalizable (hint: show that any vector x is a linear combination of eigenvectors by considering (1+aT+(aT)^2)x, where a is an eigenvalue of T). (iii) Let now T be an n x n matrix with T^3=I, and with real matrix entries. Show that tr(T) is an integer (hint: t+t^2=-1).

(b) Let H(f)=x^2f-f'' be the harmonic oscillator, and let a(f)=xf+f', a^\dagger (f)=xf-f'. Prove the following identities (you may use the identities proved in class): (i) aa^\dagger = H +I and (ii) if H(u)=\lambda u, then a^\dagger (u) is an eigenfunction of H with eigenvalue \lambda +2.

for 2/4: p. 138: 22, p. 205: 10 (hint: a) If v is an eigenvector for AB, show that Bv is an eigenvector for BA, provided it is nonzero, b) show that if A and B are n by n matrices, then AB and BA have the same eigenvalues, c) show at the example of A and B being left and right shift on l^2, that there exist operators A and B for which AB and BA do not have the same eigenvalues), look at p. 207: 17 (need not be turned in)

1) Let L(f)=(x^2-1)f''+ 2xf', where L is defined on the inner product space V of infinitely many times differentiable functions on the interval [-1,1], with the inner product given by integrating the function f*(x)g(x) over the interval [-1,1]: (a) Compute the adjoint L^\dagger. (b) Let P_n(x) be the n-th Legendre polynomial. Using (a) and the result (5.45) in the book, compute the integral \int P*_n(x)P_m(x) dx without any calculation (and without using the results in the book), if n is not equal to m. (c) By (5.45), L(P_n)=n(n+1)P_n for P_n the n-th Legendre polynomial. Are there any other eigenfunctions of L in L^2[-1,1]? Either give an example, or give a reason that there can be no other eigenfunctions.

2) Let V=L^2[-1,1]. Let U be the subspace of polynomials of degree less or equal to 2. Find the best approximation in P of the function f defined by f(x)=0 for -1 < x < 0, and f(x)=1 for 0 < x < 1.

3) Consider the function f defined by f(x)=0 for -\pi < x less equal 0, and f(x) = 1 for 0 < x < \pi. a) Calculate the integral \int f'g dx from -\pi to \pi, where g is differentiable, using integration by parts. Give an interpretation of f'. b) Calculate the Fourier series of f and f'. c) Calculate the values of the Fourier series of f at 0 (optional), and also calculate the series \sum_n cos(2n+1)x for x not equal to 0, and with n going from 0 to infinity.

for 2/18: p. 296: 7, 19, 20, p. 462: 13 and

Consider the differential operator L[y]= (1-x^2)y'' -2(m+1)xy'. (a) Find w(x) such that L is a Hermitian operator with respect to the inner product (f,g)=\int_{-1}^1 f^*(x)g(x)w(x) dx. (b) Let P_n be a polynomial of degree n which is an eigenfunction of L. Compute its eigenvalue. (c) Let Q_n be the eigenfunction given by the generalized Rodrigues formula (see book, equ (5.126), with K_n=1). Show that (xQ_n, Q_k) = 0 for k < n-1, with the inner product as in (a). (Hint: Use integration by parts, differentiating xQ_k sufficiently many times; see also the proof of theorem 5.3) (d) Deduce from this that there exist numbers a_n, b_n, c_n such that Q_{n+1}=a_n xQ_n + b_nQ_n + c_n Q_{n-1}.

for 3/4: p. 511: 4b, p. 514: 16, 18

1. Let $A$ be an n by n matrix, acting on the n-dimensional complex vector space C^n with the usual complex inner product. Show: (a) If A is self-adjoint, then the norm || A || is equal to the absolute value of its largest eigenvalue, (b) Show that || A ||^2 = || A*A || for any arbitrary matrix, (c) give an example for which || A || is strictly larger than the largest eigenvalue of A.

2) This problem gives an example of a selfadjoint operator with uncountably many spectral values, but no eigenvalues. Let V = L^2[0,1] and let X be the linear operator on V defined by (Xf)(x)=xf(x). (i) Show that X is self-adjoint (ii) Determine the norm || X ||. (iii) Show that (X-tI) is not invertible for any number t in [0,1]. (hint: It suffices to show that there can not be any bounded linear operator B on V such that B(X-tI)(f)=f for all f in V; show that if such a B existed, its norm would have to become arbitrarily large by finding functions f for which the norm of (X-tI)f becomes very small). (iv) Show that there exists no L^2 function f which would be an eigenfunction of X.

additional exercises p. 574: 12 and

Consider the operator A given by the kernel k(x,y)=x+y on the interval [0,1]. Write A=A_0 + A_1, where A_0 is the operator given by the kernel k_2 obtained by approximating k via step functions for corrresponding to the four subsquares of area 1/4 of [0,1]x[0,1]. Calculate the eigenvalues of A_0 and eigenfunctions, and estimate the norm of A_1.

Preparation for final: The exam problems will be similar to homework problems or to problems from previous exams. As examples, you can have a look at the midterm and final I gave four years ago. In that year, I did less on Sturm-Liouville functions than this year. You can also have a look at the problems from previous courses, and the suggested problems for preparation for the final there. Office hour: Monday 11-12.

## MATERIAL FROM PREVIOUS COURSE: PLEASE IGNORE

for 2/21 (Preparation for midterm - need not be turned in)

c) Find the line which best approximates the points (0,0), (1,3), (2,1), (3,-3). Find similarly the best parabola approximation. Hint: If a_0, a_1, a_2,... are the coefficients of the polynomial which approximates the points (x_i,y_i), we have the equations (y,x^(k)) = \sum_j (x^(k),x^(j)) a_j for k=0,1,... , where x^(k) is the vector whose i-th coordinate is equal to x_i^k. (I decided to use round brackets for the scalar products here as the formulas look more confusing with < > brackets in html).

for 3/7: p. 296: 6

a) Let the Fourier coefficients a_n and b_n be defined as in the book in (5.52) on page 241. Show that < f,f >= \pi(2|a_0|^2+ \sum |a_n|^2+ |b_n|^2). Moreover, show that if f is differentiable with continuous derivative then also \sum n(|a_n|^2+|b_n|^2) converges (hint: compute Fourier coefficients of f').

b) Let T : V -> V be a bounded operator on a dense subspace V of a Hilbert space H. Show that T can be extended to a unique bounded linear operator on H.

c) Compute the Fourier transform \hat y_n of the function y_n(x)= H_n(x)e^{-x^2/2}; here H_n is the n-th Hermite polynomial. The only things you need to know about Hermite polynomials is that H_0=1, H_{n+1}=2xH_n-H'_n and H_n'=2nH_{n-1} (for part c2).

c1) Show that \hat f'(x)=ix\hat f(x) c2) Show that 2y'_n=-y_{n+1}+2ny_{n-1} c3) Compute the Fourier transform of y_n by induction on n (you already know the transform for y_0, and you can compute it easily for y_1: guess from this what the general solution should be).

Preparation for final: p. 459: 1, 2, 13 p. 511: 4, 18, p. 571: 4, 12,

Prove that the integral operator T_K on L^2(I) (I an interval) is Hermitian if K(y,x) is equal to the complex conjugate of K(x,y) for all x,y in I.