Inverse scattering problem with fixed energy and fixed 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 $q$.
 Publication:

arXiv eprints
 Pub Date:
 June 2006
 arXiv:
 arXiv:mathph/0606055
 Bibcode:
 2006math.ph...6055R
 Keywords:

 Mathematical Physics;
 35J05;
 35J10;
 35R30;
 74J25;
 81U40;
 81V05