Three-dimensional inverse acoustic scattering problem by the BC-method

M. I. Belishev, A. F. Vakulenko

Published 2024-07-29Version 1

The forward acoustic scattering problem that we deal with, is to find $u=u^f(x,t)$ satisfying \begin{align*} &u_{tt}-\Delta u+qu=0, && (x,t) \in {\mathbb R}^3 \times (-\infty,\infty); \\ &u \mid_{|x|<-t} =0 , && t<0;\\ &\lim_{s \to -\infty} s\,u((-s+\tau)\,\omega,s)=f(\tau,\omega), && (\tau,\omega) \in \Sigma:=[0,\infty)\times S^2; \end{align*} for a real valued compactly supported potential $q=q(x)$ and a control $f \in L_2(\Sigma)$. The map $R: L_2(\Sigma)\to L_2(\Sigma)$, \begin{align*} & (Rf)(\tau ,\omega )\,:= \lim_{s \to +\infty} s\, u^f((s+\tau )\,\omega ,s), \quad (\tau ,\omega ) \in \Sigma \end{align*} is a response operator. The inverse problem is to determine $q$ from $R$. It is solved by a relevant version of the boundary control method. The procedure that recovers the potential is local: for any fixed $\xi>0$, given $R\upharpoonright\{f\in L_2(\Sigma)\,|\,\,f\mid_{0\leqslant \tau<\xi}=0\}$ it determines $q\big|_{|x|\geqslant\xi}$.