We study the two-dimensional Dirac operator with a class of interface conditions along a smooth closed curve, which model the so-called electrostatic and Lorentz scalar interactions of constant strengths, and we provide a rigorous description of their self-adjoint realizations and their qualitative spectral properties. We are able to cover in a uniform way all so-called critical combinations of coupling constants, for which there is a loss of regularity in the operator domain. For the case of a non-zero mass term, this results in an additional point in the essential spectrum, which reflects the creation of an infinite number of eigenvalues in the central gap, and the position of this point can be made arbitrary by a suitable choice of the parameters. 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.
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