Inverse scattering with the data at fixed energy and fixed incident direction

A. G. Ramm

paper

Inverse scattering with the data at fixed energy and fixed incident direction

arXiv:1302.5000

Abstract

Consider the Schrödinger operator $- \nabla^{2} + q$ q $,$ q=q(x), x \in \mathbf{R}^3 $. L e t$ A(β,α, k) $b e t h ecor r es p o n d in g sc a tt er in g am pl i t u d e,$ k^2 $b e t h ee n er g y,$ α\in S^2 $b e t h e in c i d e n t d i r ec t i o n,$ β\in S^2 $b e t h e d i r ec t i o n o f sc a tt er e d w a v e,$ S^2 $b e t h e u ni t s p h er e in$ \mathbf{R}^3 $. A ss u m e t ha t$ k=k_0 >0 $i s f i x e d, an d$ α=α_0 $i s f i x e d . T h e n t h esc a tt er in g d a t aa r e$ A(β)= A(β,α_0, k_0)=A_q(β) $i s a f u n c t i o n o n$ S^2 $. T h e f o l l o w in g in v er s$ \textit{IP: Given an arbitrary $f \in L^{2} (S^{2})$ and an arbitrary small number $$ $q \in C_{0}^{\infty} (D)$ , where $D \in R^{3}$ is an arbitrary fixed domai||A_q(β)-f(β)||_{L^2(S^2)} < ε$?} A positive answer to this question is given. A method for constructing such a $q $i s p r o p ose d . T h er e a r e in f ini t e l y man y s u c h$ q$, not necessarily real-valued.