Self-adjointness of two dimensional Dirac operators on corner domains

We study the self-adjointenss of the two-dimensional Dirac operator 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 H1/2, the formal form domain of the free Dirac operator. The main part of our paper consists of 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.

[1]  S. Blundell,et al.  The Dirac Equation , 2014 .

[2]  L. Vega,et al.  A strategy for self-adjointness of Dirac operators: Applications to the MIT bag model and $\delta$-shell interactions , 2016, Publicacions Matemàtiques.

[3]  L. Treust,et al.  Self-Adjointness of Dirac Operators with Infinite Mass Boundary Conditions in Sectors , 2017, 1707.04000.

[4]  On Dirac operators with electrostatic $\delta$-shell interactions of critical strength , 2016, Journal of Spectral Theory.

[5]  S. Fournais,et al.  Self-Adjointness of Two-Dimensional Dirac Operators on Domains , 2017, 1704.06106.

[6]  G. Burton.  Sobolev Spaces , 2013 .

[7]  M. Holzmann,et al.  Dirac operators with Lorentz scalar shell interactions , 2017, Reviews in Mathematical Physics.

[8]  The relativistic spherical δ -shell interaction in R 3 : Spectrum and approximation , 2017, 1704.07605.

[9]  J. Behrndt,et al.  ON DIRAC OPERATORS WITH ELECTROSTATIC δ-SHELL INTERACTIONS OF CRITICAL STRENGTH , 2017 .

[10]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[11]  C. Pommerenke.  Boundary Behaviour of Conformal Maps , 1992 .

[12]  T. Ourmières-Bonafos,et al.  Dirac Operators and Shell Interactions: A Survey , 2019, Springer INdAM Series.

[13]  M. I. Katsnelson,et al.  Chaotic Dirac Billiard in Graphene Quantum Dots , 2007, Science.

[14]  E. Stockmeyer,et al.  Infinite mass boundary conditions for Dirac operators , 2016, Journal of Spectral Theory.

[15]  Luis Vega,et al.  Shell Interactions for Dirac Operators: On the Point Spectrum and the Confinement , 2014, SIAM J. Math. Anal..

[16]  L. Vega,et al.  Shell interactions for Dirac operators , 2013, 1303.2519.

[17]  A. Mas,et al.  Klein’s paradox and the relativistic δ-shell interaction in ℝ3 , 2016, 1611.09271.

[18]  P. Seba,et al.  Dirac operators with a spherically symmetric δ-shell interaction , 1989 .

[19]  M. Berry,et al.  Neutrino billiards: time-reversal symmetry-breaking without magnetic fields , 1987, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[20]  P. Exner,et al.  On the spectral properties of Dirac operators with electrostatic δ-shell interactions , 2016, 1609.00608.

[21]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[22]  An Isoperimetric-Type Inequality for Electrostatic Shell Interactions for Dirac Operators , 2015, 1504.04220.

[23]  Kazuaki Taira.  Analytic Semigroups and Semilinear Initial Boundary Value Problems , 1995 .