An Inverse Problem for Point Inhomogeneities

We study quantum scattering theory off n point inhomogeneities (n ∈ N) in three dimensions. The inhomogeneities (or generalized point interactions) positioned at {ξ1, . . . , ξn} ⊂ R3 are modeled in terms of the n2 (real) parameter family of self-adjoint extensions of−∆ ∣∣ C∞ 0 (R3\{ξ1,...,ξn}) in L2(R3). 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.