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\begin{document}

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

\section*{Mat-1.198 Scattering Theory}

\subsection*{7$^\textrm{th}$ set of exercises, 26.3.2003}

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% \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
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% your solution on the blackboard.
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\begin{enumerate}

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\item Let $A(k,\widehat x,\alpha)$, $k>0$ being the wave number,
be the scattering amplitude for
potential scattering,
\begin{equation}\label{A}
 A(k,\widehat x,\alpha)=\int_D e^{-ik\widehat x\cdot y}q(y)u(y,k,\alpha)dy,
\end{equation}
where $u(y,k,\alpha)$ is the solution of the Lippmann-Schwinger equation,
\[
 u(x,k,\alpha)=e^{ik\alpha\cdot x}-\int_D \Phi(x-y,k)q(y)u(y,k,\alpha) dy.
\]
We can extend the wave number to negative values, too, by substituting
$-k$ to the Lippmann-Schwinger equation. Here, we assume that $q$ is
real valued and if it depends on $k$ as in the acoustic scattering,
it is an even function of $k$. The corresponding solution
satisfies no longer the outgoing radiation condition. However, it allows
us to extend $A$ to negative wave numbers, too, by the formula (\ref{A})
above.

Prove that for this extension, we have
\[
 A(-k,\widehat x,\alpha)=\overline{A(k,\widehat x,\alpha)}
=A(-k,-\alpha,-\widehat x).
\]

You may assume that $n=3$.

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\item The Born approximation for the scattering amplitude in 
material scattering is
\[
 A(k,\widehat x,\alpha)\approx A_{\rm B}(k,\widehat x,\alpha)
=\int_D e^{-ik(\widehat x-\alpha)\cdot y}q(y)dy,
\]i.e., $A_{\rm B}$ is the Fourier transform of $q$ evaluated at
\[
 \xi = k(\widehat x-\alpha)\in\R^n.
\]
Assume that the potential $q$ is independent of the wave number $k$.
(This assumption is valid in quantum mechanical scattering.) Then
$q$ itself can be reconstructed approximately by applying the inverse Fourier
transform on $A$. The problem is how to integrate $A$ that is a function
defined on $\R_+\times S^{n-1}\times S^{n-1}$. 

Prove the following integration identities in the case $n=3$:
\begin{eqnarray*}
 \int_{\R^3} f(\xi)d\xi &=& \frac 1 2\int_{-\infty}^\infty \int_{S^2}
 f(k(\widehat x-\alpha))|\widehat x -\alpha|^2 d\alpha k^2 dk \\
\noalign{\vskip4pt}
&=& \frac 1 {8\pi}\int_{-\infty}^\infty \int_{S^2}\int_{S^2}
 f(k(\widehat x-\alpha))|\widehat x -\alpha|^2d\alpha d\widehat x k^2 dk.
\end{eqnarray*}
Hence, we obtain an approximation for $q$ as
\[
 q(x)\approx q_{\rm B}(x) =\left(\frac 1{2\pi}\right)^3
 \frac 1 {8\pi}\int_{-\infty}^\infty \int_{S^2}\int_{S^2}
 e^{ik(\widehat x-\alpha)\cdot x}
A(k,\widehat x,\alpha)|\widehat x -\alpha|^2d\alpha d\widehat x k^2 dk.
\]

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(Hint: Let $\widehat x$ be a fixed parameter. If $\xi=k(\widehat x-\alpha)$,
show that
\[
 k = \frac{|\xi|}{2\widehat x\cdot\widehat\xi},\quad \alpha =\widehat x
-2(\widehat x\cdot\widehat\xi)\widehat\xi,
\]
where $\widehat\xi =\xi/|\xi|$. These formulas can be used to calculate
the Jacobian.)


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\item What are the corresponding formulas of the previous problem
in $\R^2$?

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