Abstract 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.

中文翻译：

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在闭合曲线上支持奇异相互作用的二维狄拉克算子

摘要 我们研究了具有一类沿平滑闭合曲线的界面条件的二维狄拉克算子，该算子模拟了恒定强度的所谓静电和洛伦兹标量相互作用，并严格描述了它们的自伴随实现及其定性光谱特性。我们能够以统一的方式覆盖所有所谓的耦合常数的临界组合，对于这些组合，在算子域中会失去规律性。对于非零质量项的情况，这会在基本谱中产生一个附加点，这反映了在中心间隙中产生了无数个特征值，并且该点的位置可以通过适当的参数的选择。