Inverse scattering problem with fixed energy and fixed incident direction

Details Inverse scattering problem with fixed energy and fixed incident direction Inverse scattering problem with fixed energy and fixed incident direction

2006-06-21

arxiv.org/abs/math-ph/0606055v3

Physics

Mathematical Physics

Scientific paper

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 $q$.

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