INTERACTION PROBLEMS ON PERIODIC HYPERSURFACES FOR DIRAC OPERATORS ON $$\mathbb {R}^{n}$$

Abstract

We consider the Dirac operators with singular potentials $$\begin{aligned} D_{\varvec{A},\Phi ,m,\Gamma \delta _{\Sigma }}=\mathfrak {D}_{\varvec{A},\Phi ,m}+\Gamma \delta _{\Sigma } \end{aligned}$$ (1) where $$\begin{aligned} \mathfrak {D}_{\varvec{A},\Phi ,m}= \sum \limits _{j=1}^{n} \alpha _{j}\left( -i\partial _{x_{j}}+A_{j}\right) +\alpha _{n+1}m+\Phi I_{N} \end{aligned}$$ (2) is a Dirac operator on \(\mathbb {R}^{n}\) with variable magnetic and electrostatic potentials \(\varvec{A=}(A_{1},...,A_{n}),\) \(\Phi\), and the variable mass m. In formula (2), \(\alpha _{j}\) are the \(N\times N\) Dirac matrices, that is \(\alpha _{j}\alpha _{k}+\alpha _{k}\alpha _{j}=2\delta _{jk}I_{N}\), \(I_{N}\) is the unit \(N\times N\) matrix, \(N=2^{\left[ \left( n+1\right) /2\right] },\) \(\Gamma \delta _{\Sigma }\) is a singular delta-potential supported on \(C^{2}-\)hypersurface \(\Sigma \subset \mathbb {R}^{n}\) periodic with respect to the action of a lattice \(\mathbb {G}\) on \(\mathbb {R}^{n}.\) We consider the self-adjointnes and discretness of the spectrum of unbounded in \(L^{2}(\mathbb {T},\mathbb {C}^{N})\) operators associated with the formal Dirac operator (1) on the torus \(\mathbb {T=R}^{N}\diagup \mathbb {G}\). We study the band-gap structure of the spectrum of self-adjoint operators \(\mathcal {D}\) in \(L^{2}(\mathbb {R}^{n},\mathbb {C}^{N})\) associated with the formal Dirac operator (1) on \(\mathbb {R}^{n}\) with \(\mathbb {G}\)-periodic regular and singular potentials. We also consider the Fredholm property and the essential spectrum of unbounded operators associated with non-periodic regular and singular potentials supported on \(\mathbb {G}\)-periodic smooth hypersurfaces in \(\mathbb {R}^{n}.\)