On the self-adjointness of two-dimensional relativistic shell interactions

TL;DR

This paper investigates the self-adjointness of a two-dimensional Dirac operator with shell interactions, providing explicit parameter ranges and analyzing effects of geometric features like corners, using advanced integral operator techniques.

Contribution

It introduces new explicit ranges of interaction strengths ensuring self-adjointness and employs novel analytical tools for non-smooth geometries.

Findings

Identified new parameter ranges for self-adjointness.

Analyzed effects of corner angles on operator properties.

Applied advanced integral operator methods to non-smooth curves.

Abstract

We study the self-adjointness of the two-dimensional Dirac operator coupled with electrostatic and Lorentz scalar shell interactions of constant strength $\varepsilon$ and $\mu$ supported on a closed Lipschitz curve. Namely, we present several new explicit ranges of $\varepsilon$ and $\mu$ for which there is a unique self-adjoint realization with domain included into $H^{\frac{1}{2}}$. A more precise analysis is carried out for curvilinear polygons, which allows one to take the corner openings into account. Compared to the preceding works on this topic, two new technical ingredients are employed: the explicit use of the Cauchy transform on non-smooth curves and the explicit characterization of the Fredholmness for singular integral operators.