# Two-dimensional Dirac operators with singular interactions supported on closed curves

## TL;DR

This paper rigorously analyzes the spectral properties of two-dimensional Dirac operators with singular interactions supported on closed curves, covering all coupling constants, including critical cases, and describes how their spectra can be manipulated.

## Contribution

It provides a comprehensive description of self-adjoint realizations and spectral properties of Dirac operators with singular interactions on closed curves, including critical coupling cases.

## Key findings

- Self-adjoint realizations are characterized for all coupling constants.
- Critical coupling cases lead to additional spectrum points within the gap.
- The spectral position can be controlled by adjusting coupling constants.

## Abstract

We study the two-dimensional Dirac operator with an arbitrary combination of electrostatic and Lorentz scalar $\delta$-interactions of constant strengths supported on a smooth closed curve. For any combination of the coupling constants a rigorous description of the self-adjoint realizations of the operators is given and the qualitative spectral properties are described. The analysis covers also all so-called critical combinations of coupling constants, for which there is a loss of regularity in the operator domain. In this case, if the mass is non-zero, the resulting operator has an additional point in the essential spectrum, and the position of this point inside the central gap can be made arbitrary by a suitable choice of the coupling constants. The analysis is based on a combination of the extension theory of symmetric operators with a detailed study of boundary integral operators viewed as periodic pseudodifferential operators.