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^* 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.