Self-adjointness of two-dimensional Dirac operators on 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 the adjoint operator $D^{\ast }$ 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.