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\begin{document}



Helsinki University of Technology \hfill Somersalo/Bingham \\
Mathematics

\section*{Mat-1.198 Scattering Theory}

\subsection*{4$^\textrm{th}$ set of exercises, 26.2.2003}

\vfill

% \emph{%
% By doing the exercises, you can earn points that will be taken into account 
% when determining your grade for the course:
% Do these exercises in advance, mark them as solved
% on the list at the exercise session and be prepared to present
% your solution on the blackboard.
% }

\begin{enumerate}

\vfill
\item
Consider the interior Dirichlet problem
\[
\tag{I.D.}
\begin{split}
  \left(\triangle + k^2\right) u & = 0 \qquad \text{in } D \\
  u \big|_{\partial D} & = 0
\end{split}
\]
where $D = \{ x \in \R^2 \,|\, |x| < 1 \}$.
Find a condition for the resonances $k$, i.e., for those values of $k$
for which (I.D.) has a non-trivial solution.

\vfill
\item
As Problem 1, but consider the Neumann condition,
\[
  \left.\frac{\partial u}{\partial n}\right|_{\partial D} = 0.
\]
What is the smallest resonance?
How do the resonances change when the radius of the disc changes?


\vfill
\item
By using a spherical harmonics expansion, solve the scattering problem
\[
\begin{split}
  \left(\triangle + k^2\right) u & = 0 \qquad \qquad
  \text{in } \R^2 \setminus \overline{D} = \{ x \in \R^2 \,|\, |x| > 1 \} \\
  u \big|_{\partial D} & = 0 \\
  u & = u_{\text{inc}} + u_{\text{sc}}
\end{split}
\]
where $u_{\text{sc}}$ satisfies the radiation condition and 
\[
  u_{\text{inc}}(x) = e^{i k \hat{\alpha} \cdot x},
  \qquad
  |\hat{\alpha}| = 1.
\]
\big(Hint: Use
$
  \frac 1{2\pi}\int_0^{2\pi}e^{i(z\cos\theta + n\theta)}d\theta
  =
  i^n J_n(z)
$.\big)

\vfill
\item
The integral kernel of the single layer operator on the 
unit circle was
\[
  L(t, s) = L_1(t, s) \ln \left( 4 \sin^2 \frac{t - s}{2} \right)
  + L_2(t, s),
\]
where
\[
  L_1(t, s) = - \frac{1}{4\pi}\, J_0 \left(2 k \sin \frac{t - s}{2} \right)
\]
and
\[
  L_2(t, s) = \frac{i}{4}\, H^{(1)}_0 \left(2 k \,\Big| \sin \frac{t - s}{2} 
    \, \Big| \right) - L_1(t, s) \ln \left( 4 \sin^2 \frac{t - s}{2} \right)
\]
is the regular part. We need to calculate $L_2(t, t)$.
Calculate $L_2(t, t)$ by using the asymptotics of $H^{(1)}_0$ at
the origin,
\[
  H^{(1)}_0(z) 
  = J_0(z) 
  + \frac{2i}{\pi} \left(\ln \frac{z}{2} + \gamma\right) J_0(z)
  + \sum_{k=1}^\infty a_k z^{2k}.
\]

\end{enumerate}
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