Helsinki University of Technology Somersalo/Bingham Mathematics

Mat-1.198 Scattering Theory

5^{th} set of exercises, 5.3.2003

Consider the scattering problem by a sound-hard obstacle: Given a solution u_{inc} of

(△ + k^{2})u_{inc} = 0 in ℝ^{3},

find u = u_{inc}+ u_{sc} such that

(△ + k^{2})u = 0  in ℝ^{3}\ \overline{D}
\frac{∂u}{∂n}|_{∂D} = 0
lim_{r→∞}r ( \frac{∂u_{sc}}{∂r} − iku_{sc}) = 0.

1. Find an integral equation for u|_{∂D} using the Helmholtz representation theorem.

2. Formulate the result equivalent to Theorem 11.8 (corollary of the Fredholm Alternative) for this problem.

3. Consider the previous week’s exercise concerning the scattering by a unit disc. By using Theorem 11.8 and the explicit form of the Neumann resonances in D, show that the equation

(1 + 2K′)φ = 2 \frac{∂u_{inc}}{∂n}, u_{inc}(x) = e^{ikα̂·x}

is solvable even at resonance frequencies.

4. Prove Theorem 11.2:

(1) If A: X → X is compact and B: X → X is bounded, then both AB and BA are compact.

(2) If A_{n}: X → X is compact for all n ∈ ℕ and ∥A_{n}− A∥ → 0, then A is compact.

(3) If A : X → X is bounded and finite dimensional, i.e., dimRan(A) < ∞, then A is compact.