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\begin{document}

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

\section*{Mat-1.198 Scattering Theory}

\subsection*{8$^\textrm{th}$ set of exercises, 2.4.2003}

\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}

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\item Consider the one-dimensional potential scattering problem:
Let $q:\R\to\R$ be a piecewise continuous potential, $q(x)=0$
for $|x|>M$, and $u$ a solution of the equation
\[
 u'' + k^2 u = q u.
\]
Write
\[
 u(x) = u_{\rm inc}(x) + u_{\rm sc}(x),
\]
where the incoming field is a left-moving plane wave sent from $+\infty$,
\[
 u_{\rm inc}(x)=e^{-ikx}.
\]
Part of this wave is reflected back, part is transmitted through, so that
\[
 u(x) =\left\{\begin{array}{ll}
 Re^{ikx} + e^{-ikx}, &x> M\\
 Te^{-ikx},&x<-M.
\end{array}\right.
\]
% \[
%  u_{\rm sc}(x) =\left\{\begin{array}{ll}
%  Re^{ikx}, &x> M\\
%  Te^{-ikx},&x<-M
% \end{array}\right.
% \]

(a) Write the appropriate radiation condition in this case.

(b) Derive the Lippmann-Schwinger integral equation.

(c) Find the integral representations of the reflection and transmission
coefficients $R$ and $T$.

(d) What is the Born approximations for $R$ and $T$? 

\item Prove that the scattering is unitary:
\[
 |R|^2 + |T|^2 = 1.
\]

\vfill

(Hint: Consider the Wronskian of the solutions $u(x)$ and
$\overline{u(x)}$.) 

\vfill

\item Prove the inversion formula for Abel's integral equation: If
\[
 h(y)= \int_0^y\frac {f(t)}{(y-t)^\alpha}dt,\quad 0<\alpha<1,
\]
then
\[
 f(t)=\frac{\sin\pi\alpha}{\pi}\frac{d}{dt}\int_0^t\frac {h(y)}
{(t-y)^{1-\alpha}}dy.
\]

\vfill
(Hint: Use Laplace transform and the convolution theorem.)

\vfill

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