Resonance Point Interactions

TL;DR

This paper introduces a new inverse scattering construction for point interactions and generalised point interactions, including a novel family called resonance point interactions, with detailed analysis of their mathematical and physical properties.

Contribution

It presents a new inverse scattering method for constructing point interactions, introduces a family of resonance point interactions, and analyzes their Hilbert space extensions and scattering properties.

Findings

Constructed a new class of resonance point interactions (RPI).

Extended Hilbert spaces for different RPI cases.

Confirmed scattering theory properties for the models.

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 M{\o}ller 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.