Inverse scattering problem with fixed energy and fixed incident direction

TL;DR

This paper demonstrates that for inverse scattering problems with fixed energy and incident direction, one can approximate any target scattering amplitude by a potential within a bounded domain, providing a method and formula for constructing such potentials.

Contribution

It introduces a method and explicit formula to construct potentials that approximate arbitrary scattering amplitudes in fixed energy and incident direction inverse scattering problems.

Findings

Potential functions can approximate any target scattering amplitude within a specified error.

The constructed potential is nonunique but can be explicitly found.

The method applies to potentials in bounded domains with fixed energy and incident direction.

Abstract

Let $A_q(\alpha',\alpha,k)$ be the scattering amplitude, corresponding to a local potential $q(x)$, $x\in\R^3$, $q(x)=0$ for $|x|>a$, where $a>0$ is a fixed number, $\alpha',\alpha\in S^2$ are unit vectors, $S^2$ is the unit sphere in $\R^3$, $\alpha$ is the direction of the incident wave, $k^2>0$ is the energy. We prove that given an arbitrary function $f(\alpha')\in L^2(S^2)$, an arbitrary fixed $\alpha_0\in S^2$, an arbitrary fixed $k>0$, and an arbitrary small $\ve>0$, there exists a potential $q(x)\in L^2(D)$, where $D\subset R^3$ is a bounded domain such that \bee \|A_q(\alpha',\alpha_0,k)-f(\alpha')\|_{L^2(S^2)}<\ve. \tag{$\ast$}\eee The potential $q$, for which $(\ast)$ holds, is nonunique. We give an method for finding $q$, and a formula for such a $q$.