Point Interactions from Flux Conservation

TL;DR

This paper demonstrates that flux conservation can replace traditional matching conditions in point interactions, providing a physical interpretation and extending to higher dimensions.

Contribution

It introduces a flux conservation-based approach to point interactions, generalizing the treatment of delta potentials and linking to deficiency index theory.

Findings

Flux conservation can substitute matching conditions in point interactions.

The method applies to arbitrary superpositions of delta and delta' potentials.

The approach extends naturally to higher-dimensional cases.

Abstract

We show that the physical requirement of flux conservation can substitute for the usual matching conditions in point interactions. The study covers an arbitrary superposition of $\delta$ and $\delta'$ potentials on the real line and can be easily applied to higher dimensions. Our procedure can be seen as a physical interpretation of the deficiency index of some symmetric, but not self-adjoint operators.