Self-Adjointness of two dimensional Dirac operators on corner domains

2019/01/01 by Fabio Pizzichillo, Pizzichillo, Fabio, Hanne Van Den Bosch +1 · 3 citations
Mathematics · Physics and Astronomy · #Spectral Theory in Mathematical Physics #Advanced Operator Algebra Research #Quantum Mechanics and Non-Hermitian Physics

paper · pdf · doi:10.48550/arxiv.1902.05010

Abstract

We investigate the self-adjointness of the two-dimensional Dirac operator D, with quantum-dot and Lorentz-scalar δ-shell boundary conditions, on piecewise C² 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.

Cited by

• Congruence of Dirac operators with applications to generalized MIT bag models
• MIT bag in non-smooth convex domains
• On shape optimization with large magnetic fields in two dimensions

Related

• Dirac operators with infinite mass boundary conditions on unbounded domains with infinite corners
• A strategy for self-adjointness of Dirac operators: Applications to the\n MIT bag model and delta-shell interactions
• On the self-adjointness of two-dimensional relativistic shell interactions
• Local Boundary Conditions for Dirac-type operators
• On two-dimensional Dirac operators with δ-shell interactions supported on unbounded curves with straight ends