# Self-Adjointness of two dimensional Dirac operators on corner domains

## TL;DR

This paper studies the self-adjointness of 2D Dirac operators on domains with corners, establishing unique realizations with domains in Sobolev space and analyzing explicit cases like infinite sectors.

## Contribution

It provides a comprehensive analysis of self-adjoint extensions of 2D Dirac operators on corner domains, including explicit solutions for sector geometries.

## Key findings

- Existence of a unique self-adjoint realization with domain in H^{1/2}
- Explicit characterization of self-adjoint extensions on infinite sectors
- Translation of sector results to general corner domains

## Abstract

We investigate the self-adjointness of the two-dimensional Dirac operator $D$, with quantum-dot and Lorentz-scalar $\delta$-shell boundary conditions, on piecewise $C^2$ domains with finitely many corners. For both models, we prove the existence of a unique self-adjoint realization whose domain is included in the Sobolev space $H^{1/2}$, the formal form domain of the free Dirac operator. The main part of our paper consists of a description of the domain of $D^*$ in terms of the domain of $D$ and the set of harmonic functions that verify some mixed boundary conditions. Then, we give a detailed study of the problem on an infinite sector, where explicit computations can be made: we find the self-adjoint extensions for this case. The result is then translated to general domains by a coordinate transformation.