An inverse problem for point inhomogeneities

Abstract

We study quantum scattering theory off $n$ point inhomogeneities ($n\in\bbN$) in three dimensions. The inhomogeneities (or generalized point interactions) positioned at $\{ξ_1,...,ξ_n\}\subset\bbR^3$ are modeled in terms of the $n^2$ (real) parameter family of self-adjoint extensions of $-Δ\big|_{C^\infty_0(\bbR^3\backslash\{ξ_1,...,ξ_n\})}$ in $L^2(\bbR^3)$. The Green's function, the scattering solutions and the scattering amplitude for this model are explicitly computed in terms of elementary functions. Moreover, using the connection between fixed energy quantum scattering and acoustical scattering, the following inverse spectral result in acoustics is proved: The knowledge of the scattered field on a plane outside these point-like inhomogeneities, with all inhomogeneities located on one side of the plane, uniquely determines the positions and boundary conditions associated with them.