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.
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.