Helsinki University of Technology Somersalo/Bingham Mathematics

Mat-1.198 Scattering Theory

4^{th} set of exercises, 26.2.2003

1. Consider the interior Dirichlet problem

\begin{aligned}(△ + k^{2}) u & = 0  in D \\ u|_{∂D} & = 0 \end{aligned}  (I.D.)

where D = {x ∈ ℝ^{2}||x| < 1}. Find a condition for the resonances k, i.e., for those values of k for which (I.D.) has a non-trivial solution.

2. As Problem 1, but consider the Neumann condition,

\frac{∂u}{∂n}|_{∂D} = 0.

What is the smallest resonance? How do the resonances change when the radius of the disc changes?

3. By using a spherical harmonics expansion, solve the scattering problem

\begin{aligned}(△ + k^{2}) u & = 0 in ℝ^{2}\ \overline{D}= \{x ∈ ℝ^{2}||x| > 1\} \\ u|_{∂D} & = 0 \\ u & = u_{inc}+ u_{sc} \end{aligned}

where u_{sc} satisfies the radiation condition and

u_{inc}(x) = e^{ikα̂·x}, |α̂| = 1.

(Hint: Use \frac{1}{2π} ∫_{0}^{2π}e^{i(zcosθ+nθ)}dθ = i^{n}J_{n}(z).)

4. The integral kernel of the single layer operator on the unit circle was

L(t,s) = L_{1}(t,s)ln( 4sin^{2} \frac{t − s}{2} ) + L_{2}(t,s),

where

L_{1}(t,s) = −\frac{1}{4π} J_{0}( 2ksin \frac{t − s}{2} )

and

L_{2}(t,s) = \frac{i}{4} H_{0}^{(1)}( 2k|sin \frac{t − s}{2} |) − L_{1}(t,s)ln( 4sin^{2} \frac{t − s}{2} )

is the regular part. We need to calculate L_{2}(t,t). Calculate L_{2}(t,t) by using the asymptotics of H_{0}^{(1)} at the origin,

H_{0}^{(1)}(z) = J_{0}(z) + \frac{2i}{π} ( ln \frac{z}{2} + γ) J_{0}(z) + ∑_{k=1}^{∞} a_{k}z^{2k}.