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

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