Mathematical Methods in Physics and Engineering, Winter quarter 2001:

Office hours: MW: 2-3, 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 50% Midterm 25% Homework 25%

Dates of exams: Midterm: February 21, Final: March 20, 3-6 in usual class room.

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 1/24: (you are allowed to use properties of determinants stated in the book) p. 136: 3, 4, 10, 11(A is antisymmetric if its transposed is equal to -A), 14, 17 and

a) Let V be a finite dimensional vector space with basis B = {x_1, ..., x_n}, and let S,T be linear transformations of V represented by the matrices A and C with respect to the basis B. Show that the transformation ST, which maps the vector x to the vector S(Tx) is represented by the matrix AC (usual matrix multiplication).

b) 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).

for 2/7: p. 138: 22, p. 205: 1, 7, 9, 10(only for special case A, B being n x n matrices) and

a) Let V be the vector space of differentiable functions on the interval [0,1], and let the inner product < f,g > be defined by integrating the function \bar f(x) g(x) over [0,1]. Show that the adjoint of D_x: f -> f' is not equal to -D_x.

b) Let A: V -> V be linear such that A^\dagger A= AA^\dagger, i.e. A commutes with its adjoint. Show that A can be diagonalized (V is finite dimensional).

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

a) 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.

b) Let V be as in a), and let X be the linear operator on V defined by (Xf)(x)=xf(x). (i) Show that X is self-adjoint (ii) Find numbers a and b such that a \leq < Xf,f> \leq b for all f in V with ||f||=1 ("\leq" means less or equal). (iii) Find the best possible numbers a and b in (ii), i.e. show for any c>a there is a function f with ||f||=1 such that < c. (iv) Show that (X-tI) is not invertible for any number t in [a,b] (hint: compute (X-tI)(f)(t) - to keep things simpler you can assume V to be the space of continuous functions on I for this part of the problem)

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.