Generalized point interactions for the radial Schrodinger equation via unitary dilations

Abstract

We present an inverse scattering construction of generalised point interactions (GPI) -- point-like objects with non-trivial scattering behaviour. The construction is developed for single centre $S$-wave GPI models with rational $S$-matrices, and starts from an integral transform suggested by the scattering data. The theory of unitary dilations is then applied to construct a unitary mapping between Pontryagin spaces which extend the usual position and momentum Hilbert spaces. The GPI Hamiltonian is defined as a multiplication operator on the momentum Pontryagin space and its free parameters are fixed by a physical locality requirement. We determine the spectral properties and domain of the Hamiltonian in general, and construct the resolvent and M{\o}ller wave operators thus verifying that the Hamiltonian exhibits the required scattering behaviour. The physical Hilbert space is identified. The construction is illustrated by GPI models representing the effective range approximation. For negative effective range we recover a known class of GPI models, whilst the positive effective range models appear to be new. We discuss the interpretation of these models, along with possible extensions to our construction.