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 $D^*$ 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.
- Publication:
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arXiv e-prints
- Pub Date:
- February 2019
- DOI:
- 10.48550/arXiv.1902.05010
- arXiv:
- arXiv:1902.05010
- Bibcode:
- 2019arXiv190205010P
- Keywords:
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- Mathematics - Analysis of PDEs;
- Mathematical Physics