An inverse problem for point inhomogeneities (Q2711778)

Let \(\Delta_{\theta,\Xi}\) be the operator in \(L_2(\mathbb R^3)\) generated by the Laplacian and the point interaction concentrated on a finite set \(\Xi =\{ \xi_1,\ldots \xi_n\}\) with the selfadjoint realization parametrized by an \(n\times n\) matrix \(\theta\) characterizing the boundary conditions. It is shown that the Green function \(G(k^2,x,y)\) of the operator \(\Delta_{\theta,\Xi}\) for a fixed value of the spectral parameter \(k^2>0\) and \(x\neq y\) from some plane uniquely determine the set \(\Xi\) and the matrix \(\theta\). In terms of the scattering theory this means that the positions of point inhomogeneities located on one side of a plane as well as the boundary conditions are determined by the scattered field on a plane.