Interaction of Relativistic Particles with Singular Potentials Supported by a Periodic Graph

We consider the interaction of relativistic particles described by two-dimensional Dirac operators with delta-type singular potentials supported by periodic graphs \(\Gamma\subset\mathbb{R}^{2}\). This problem can be regarded as a relativistic analog of the Kronig–Penney model of electron propagation in solid state physics. We associate with this problem an unbounded operator in the Hilbert space \(L^{2}(\mathbb{R}^{2},\mathbb{C}^{2})\). The study of spectral properties of these operators is reduced to the study of the Fredholmness of singular integral operators on the graph \(\Gamma\). We obtain necessary and sufficient conditions for the Fredholmness of these operators as ellipticity conditions on the edges, matrix conditions at the vertices, and conditions of invertibility of limit operators which are periodic operators on the graph \(\Gamma\). We apply the Bloch–Floquet theory to the study of invertibility of limit operators.

DOI 10.1134/S1061920825600163