Math 642-600 Assignments Spring 2007
Assignment 1 - Due Thursday, February 1.
- Do the following problems.
- Problem 2, page 204 (§ 5.1).
- Problem 6, page 204 (§ 5.1).
- Problem 7, page 204 (§ 5.1).
- Hanging chain problem. A chain having uniform linear density
ρ hangs between the points (0,0) and (1,0). The total mass m,
which is fixed, and the total energy E of the chain are
Assuming that the chain hangs in a shape that minimizes the
energy, find the shape of the hanging chain.
- Consider the functional J(y) =
∫abF(x,y,y′)dx, where y ∈
C1 and y(a)=A, y(b)=B are fixed.
- Let y extremize J, subject to all of the conditions above. Verify
that if η ∈ C1 and η(a)=η(b)=0,
then
Δ J = ∫ab (-g(x) +
Fy′(x,y,y′))η′(x)dx, where g(x) =
∫axFy(u,y,y′)du.
- Use (a) to show that -g+Fy′(x,y,y′)=c, a
constant. From this, it follows that
Fy′(x,y,y′)∈ C1. (Du
Bois-Reymond, 1879.)
Assignment 2 - Due Tuesday, February 13.
- Read sections 5.4 and 6.1.
- Do the following problems.
- Problem 4, page 205 (§ 5.2).
- Problem 6, page 205 (§ 5.2).
- Problem 7, page 206 (§ 5.2).
- Problem 8, page 206 (§ 5.2). Also, find the Hamiltonian of
the system, and write out Hamilton's equations for it. Finding
integrals (constants) of the motion should follow immediately from
these equations.
- Find the Legendre transformation H(p) for F(x) =
xT A x, where x is in
Rn and A is a symmetric, positive definite n×n matrix.
Assignment 3 - Due Tuesday, February 20.
- Read sections 6.1, 6.2, and 6.4.
- Do the following problems.
- Problem 1, page 207-8 (§ 5.4).
- Problem 4, page 208 (§ 5.4).
- Problem 6, page 208 (§ 5.4). Work with n = 2, and Ω a
disk centered at 0 and having radius r = a. How does the lowest eigenvalue
change with the radius a?
- Problem 8, page 208 (§ 5.4).
- Problem 4, page 274 (§ 6.1).
- Problem 6, page 276 (§ 6.2).
Assignment 4 - Due Thursday, March 1.
- Read section 6.5.
- Do the following problems.
- Problem 2, page 277-278 (§ 6.4)
- Problem 4, page 277-278 (§ 6.4)
- Problem 17, page 277-278 (§ 6.4)
- Problem 25, page 277-278 (§ 6.4)
- Problem 26, page 277-278 (§ 6.4)
Assignment 5 - Due Tuesday, April 2.
- Read section 7.1 and 7.2.1.
- Do the following problems.
- Define the following terms. (All refer to linear operators on a Hilbert space H.) You may have to look up some terms in texts or on the web.
- Adjoint of L
- Two operators L1 and L2 are equal
- L2 is an extension of L1
- L is densely defined
- L is closed
- L is closable
- Resolvent set
- Spectrum -- pure point, residual, and continuous
- Graph of an operator
- Show that if L is self adjoint, then its spectrum is a subset of
the reals.
- Let L = - i d/dx and DL = {u in L2(R) | u'
is in L2(R)}. Show that L is self adjoint.
- For the operator L in the previous problem, show that 0 belongs
to the continuous spectrum.
- Problem 2(a,b), page 328 (§ 7.1)
Assignment 6 - Due Friday, April 27.
- Read sections 10.1-10.5.
- Do the following problems.
- Problem 1(b), page 329 (§ 7.2)
- Problem 2, page 329 (§ 7.2)
- Let q(x) ≥ 0 be a compactly supported, bounded
(L∞), potential on R. Let Hu= -u′′
+ q(x)u, be a Schrödinger operator, which you are given to be
selfadjoint on all u such that u′′ is in L2.
- Show that there is no point spectrum.
- Show that the spectrum is the interval [0,∞). (Hint: for
λ to be in the reolvent set, given v=(H-λI)u, we have to
have ||u|| ≤ C||v||. Show that this is possible for λ <
0, but impossible for λ ≥ 0.)
- Problem 3(d), page 333 (§ 7.5)
- For the delta function potential we did in class (see notes for
12 April), find the spectral transform for f(x) = H(x+2)-H(x+1). (This
a step that is 1 on -2 < x < -1 and 0 elsewhere.)
- Let I0(x) be the modified Bessel function of
the first kind. Using the integral formula
I0(x) = (1/π)∫0π cosh(x
cos(t))dt,
derive the asymptotic formula on the top of p. 266 in the text.
Updated 4/20/07 (fjn).