INTERACTION PROBLEMS ON PERIODIC HYPERSURFACES FOR DIRAC OPERATORS ON Rn.

We consider the Dirac operators with singular potentials 1 D A , Φ , m , Γ δ Σ = D A , Φ , m + Γ δ Σ where 2 D A , Φ , m = ∑ j = 1 n α j - i ∂ x j + A j + α n + 1 m + Φ I N is a Dirac operator on R n with variable magnetic and electrostatic potentials A = (A 1 ,... , A n) , Φ , and the variable mass m. In formula (2), α j are the N × N Dirac matrices, that is α j α k + α k α j = 2 δ jk I N , I N is the unit N × N matrix, N = 2 n + 1 / 2 , Γ δ Σ is a singular delta-potential supported on C 2 - hypersurface Σ ⊂ R n periodic with respect to the action of a lattice G on R n. We consider the self-adjointnes and discretness of the spectrum of unbounded in L 2 (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 L 2 (R n , C N) associated with the formal Dirac operator (1) on R n 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 R n. [ABSTRACT FROM AUTHOR]