Resonance Point Interactions

Abstract

A new construction is presented for point interactions (PI) and generalised point interactions (GPI). The construction is an inverse scattering procedure, using integral transforms suggested by the required scattering theory. The usual class of PI in 3 dimensions (i.e. the self adjoint extensions of the Laplacian on the domain of smooth functions compactly supported away from the origin) is reconstructed. In addition a 1-parameter family of GPI models termed resonance point interactions (RPI) is constructed, labelled by $M$. The case $M<0$ coincides with a special case of a known GPI model; the case $M>0$ appears to be new. In both cases, the Hilbert space of states must be extended, for $M<0$, a larger Hilbert space is required, whilst for $M>0$, the Hilbert space is extended to a Pontryagin space. In the latter case, the space of physical states is identified as a positive definite invariant subspace. Complete Møller wave operators are constructed for the models considered, using a two space formalism where necessary, which confirm that the PI and RPI models exhibit the required scattering theory. The physical interpretation of RPI as models for quantum mechanical systems exhibiting zero energy resonances is described.