INTERACTION PROBLEMS ON PERIODIC HYPERSURFACES FOR DIRAC OPERATORS ON Rⁿ

We consider the Dirac operators with singular potentials DA,Φ,m,ΓδΣ=DA,Φ,m+ΓδΣwhere DA,Φ,m=∑j=1nαj(-i∂xj+Aj)+αn+1m+ΦINis a Dirac operator on Rⁿ with variable magnetic and electrostatic potentials A=(A₁,.. , A_n) , Φ , and the variable mass m. In formula (2), α_j are the N× N Dirac matrices, that is α_jα_k+ α_kα_j= 2 δ_jkI_N, I_N is the unit N× N matrix, N= 2 ^[(n+1)/2], Γ δ_Σ is a singular delta-potential supported on C²- hypersurface Σ ⊂ Rⁿ periodic with respect to the action of a lattice G on Rⁿ. We consider the self-adjointnes and discretness of the spectrum of unbounded in L²(T, C^N) operators associated with the formal Dirac operator (1) on the torus T= R^N∕G. We study the band-gap structure of the spectrum of self-adjoint operators D in L²(Rⁿ, C^N) associated with the formal Dirac operator (1) on Rⁿ with 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 G-periodic smooth hypersurfaces in Rⁿ.