Non-self-adjoint relativistic point interaction in one dimension

TL;DR

This paper introduces a non-self-adjoint one-dimensional Dirac operator with a singular interaction, analyzes its spectral properties, explores its non-relativistic limit, and investigates regular approximations, highlighting differences between local and non-local cases.

Contribution

It presents a novel non-self-adjoint Dirac operator with singular interactions, studying its spectral behavior and limits, and clarifies the effects of local versus non-local regularizations.

Findings

Spectral properties of the operator are characterized.

Non-relativistic limit of the operator is derived.

Coupling constants are not renormalized for non-local approximations.

Abstract

The one-dimensional Dirac operator with a singular interaction term which is formally given by $A\otimes|\delta_0\rangle\langle\delta_0|$, where $A$ is an arbitrary $2\times 2$ matrix and $\delta_0$ stands for the Dirac distribution, is introduced as a closed not necessarily self-adjoint operator. We study its spectral properties, find its non-relativistic limit and also address the question of regular approximations. In particular, we show that, contrary to the case of local approximations, for non-local approximating potentials, coupling constants are not renormalized in the limit.