Uniqueness of the multi-dimensional inverse scattering problem for time dependent potentials
We prove the uniqueness of the inverse scattering problem for the wave equation \(u_{tt}-\Delta u+q(t,x)u=0,\) \(t\in {\mathbb{R}}\), \(x\in {\mathbb{R}}^ n\), \(n\geq 3\) odd with time dependent potential q(t,x). The function q(t,x) is assumed to be smooth, compactly supported in x and having polynomial growth in t. We introduce the generalized scattering kernel \(K^{\#}(s',\omega ';s,\omega)\in C^{\infty}(S_{\omega '}^{n- 1}\times {\mathbb{R}}_ s\times S_{\omega}^{n-1};\quad {\mathcal D}'({\mathbb{R}}_{s'}))\) which describes the scattering of the plane waves \(\delta (t+s-x\cdot \omega)\). If the scattering operator S exists, \(K^{\#}\) is the kernel of S-I. Our main result is the following uniqueness theorem. Let \(K_ i^{\#}\) be related to \(q_ i\), \(i=1,2\) and suppose that \(K_ 1^{\#}=K_ 2^{\#}\) for \(| \omega - \omega_ 0| <\epsilon\), \(| \omega '-\omega_ 0| <\epsilon\), \(| s'-s| <\epsilon\) with some \(\epsilon >0\), \(\omega_ 0\in S^{n-1}\). Then \(q_ 1=q_ 2\). The proof is completely time dependent. First we derive suitable representation of \(K^{\#}\). Then we reconstruct integrals of q(t,x) over some characteristic rays by examination of \(K^{\#}\) near the singularity \((s',\omega ')=(s,\omega)\). Finally we show that these integrals determine q(t,x) uniquely.
scientific article; zbMATH DE number 4065527
scientific article