Math 642-600 Current Assignments — Spring 2007

Assignment 6 - Due Friday, April 27.

• Read sections 10.1-10.5.
• Do the following problems.
  1. Problem 1(b), page 329 (§ 7.2)
  2. Problem 2, page 329 (§ 7.2)
  3. 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.
     1. Show that there is no point spectrum.
     2. 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.)
  4. Problem 3(d), page 333 (§ 7.5)
  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.)
  6. Let I₀(x) be the modified Bessel function of the first kind. Using the integral formula
     I₀(x) = (1/π)∫₀^π cosh(x cos(t))dt,
     derive the asymptotic formula on the top of p. 266 in the text.

Updated 4/20/07 (fjn).