Two-dimensional Dirac operators with general $\delta$-shell interactions supported on a straight line

TL;DR

This paper introduces a self-adjoint two-dimensional Dirac operator with general delta-shell interactions on a line, analyzing its spectral properties and potential as a model for Dirac materials.

Contribution

It defines a new class of Dirac operators with delta-shell interactions and thoroughly investigates their spectral characteristics and approximations.

Findings

Singularly continuous spectrum is always empty.

Switching delta-shell interactions can create eigenvalues in spectral gaps.

Operators can approximate Dirac materials with regular potentials.

Abstract

In this paper the two-dimensional Dirac operator with a general hermitian $\delta$-shell interaction supported on a straight line is introduced as a self-adjoint operator and its spectral properties are investigated in detail. In particular, it is demonstrated that the singularly continuous spectrum is always empty and that by switching a certain $\delta$-shell interaction on, it is possible to generate an eigenvalue in the gap of the spectrum of the free operator or to partially or even fully close the gap. This suggests that the studied operators may serve as interesting continuum toy-models for Dirac materials. Finally, approximations by Dirac operators with regular potentials are presented.