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\begin{document}

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

\section*{Mat-1.198 Scattering Theory}

\subsection*{5$^\textrm{th}$ set of exercises, 5.3.2003}

\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.
% }

Consider the scattering problem by a sound-hard obstacle: 
Given a solution $u_{\text{inc}}$ of
\[
  (\triangle + k^2) u_{\text{inc}} = 0
  \qquad \text{in } \R^3,
\]
find $u = u_{\text{inc}} + u_{\text{sc}}$ such that
\[
\begin{split}
  (\triangle + k^2) u & = 0 
  \qquad \text{in } \R^3 \setminus \overline{D}
  \\ 
  \left. \frac{\partial u}{\partial n} \right|_{\partial D} 
  & 
  = 0 \\
  \lim_{r \to \infty} r \left( \frac{\partial u_{\text{sc}}}{\partial r} 
    - i k u_{\text{sc}} \right) 
  & 
  = 0.
\end{split}
\]

\begin{enumerate}

\vfill
\item
Find an integral equation for $u|_{\partial D}$ using the Helmholtz 
representation theorem.


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


\vfill
\item
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
\[
  (1 + 2 K') \varphi = 2 \,\frac{\partial u_{\text{inc}}}{\partial n},
  \qquad
  u_{\text{inc}}(x) = e^{i k \hat{\alpha} \cdot x}
\]
is solvable even at resonance frequencies.


\vfill
\item
Prove \textbf{Theorem 11.2:}\emph{
  \begin{enumerate}[(1)]
  \item If $A: X \to X$ is compact and $B: X \to X$ is bounded,
    then both $AB$ and $BA$ are compact.
  \item If $A_n: X \to X$ is compact for all $n \in \N$ and
    $\|A_n - A\| \to 0$, then $A$ is compact.
  \item If $A: X \to X$ is bounded and finite dimensional, i.e.,
    $\dim \Ran(A) < \infty$, then $A$ is compact.
  \end{enumerate}
}

\vfill

\end{enumerate}
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