General $\delta$-shell interactions for the two-dimensional Dirac operator: self-adjointness and approximation

TL;DR

This paper studies the self-adjointness and spectral properties of a 2D Dirac operator with complex singular interactions supported on a curve, and how to approximate it with regular potentials, advancing understanding of such quantum systems.

Contribution

It provides a systematic analysis of general delta-shell interactions for the 2D Dirac operator, including self-adjointness, spectral description, and approximation methods.

Findings

Characterization of self-adjoint extensions for the operator.

Spectral analysis of the Dirac operator with delta-shell interactions.

Approximation of singular interactions by regular potentials.

Abstract

In this work we consider the two-dimensional Dirac operator with general local singular interactions supported on a closed curve. A systematic study of the interaction is performed by decomposing it into a linear combination of four elementary interactions: electrostatic, Lorentz scalar, magnetic, and a fourth one which can be absorbed by using unitary transformations. We address the self-adjointness and the spectral description of the underlying Dirac operator, and moreover we describe its approximation by Dirac operators with regular potentials.