Math 642-600 Current Assignments Spring 2007
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).