Two-dimensional Dirac operators with singular interactions supported on closed curves

Abstract

We study the two-dimensional Dirac operator with an arbitrary combination of electrostatic and Lorentz scalar $\delta$-interactions of constant strengths supported on a smooth closed curve. For any combination of the coupling constants a rigorous description of the self-adjoint realizations of the operators is given and the qualitative spectral properties are described. The analysis covers also all so-called critical combinations of coupling constants, for which there is a loss of regularity in the operator domain. In this case, if the mass is non-zero, the resulting operator has an additional point in the essential spectrum, and the position of this point inside the central gap can be made arbitrary by a suitable choice of the coupling constants. The analysis is based on a combination of the extension theory of symmetric operators with a detailed study of boundary integral operators viewed as periodic pseudodifferential operators.

arXiv comment54 pages. Some changes in the structure of the text

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• Spectral Analysis of Dirac Operators with delta interactions supported on the boundaries of rough domains
• A variational formulation for Dirac operators in bounded domains. Applications to spectral geometric inequalities
• Two-dimensional Dirac operators with general $\delta$-shell interactions supported on a straight line
• Eigenvalue curves for generalized MIT bag models
• Schrödinger operators with oblique transmission conditions in ${\mathbb{R}}^2$
• Curvature contribution to the essential spectrum of Dirac operators with critical shell interactions
• Limiting absorption principle and scattering matrix for Dirac operators with $\delta$-shell interactions
• On two-dimensional Dirac operators with critical delta-shell interactions