Dirac Operators with Delta-Interactions on Smooth Hypersurfaces in Rn

We consider the Dirac operators with singular potentials D-A,D-Phi,D-m,D-Gamma delta Sigma = D-A,D-Phi,D-m + Gamma delta(Sigma) (1) where D-A,D-Phi,D-m = Sigma(n)(j=1) alpha(j)(-i partial derivative(chi j) + A(j)) + alpha(n) + (1)m + Phi I-N (2) is a Dirac operator on R-n with variable magnetic and electrostatic potentials A = (A(1), ..., A(n)) is an element of L-infinity (R-n, C-n) and Phi is an element of L-infinity (R-n), and the variable mass m is an element of L-infinity (R-n). In formula (2) alpha(j) are the N x N Dirac matrices, that is alpha(j)alpha(k) + alpha k alpha(j) = 2 delta I-jk(N), I-N is the unit N x N matrix, N = 2([(n + 1)/2]). In formula (1) Gamma delta(Sigma) is a singular delta-type potential supported by a C-2-hypersurface Sigma subset of R-n which is the common boundary of the open sets Omega(+/-). Let H-1 (Omega(+/-) , C-N) be the Sobolev spaces of N-dimensional vector-valued distributions u on Omega(+/-), and H-1(R-n\Sigma,C-N) = H-1(Omega(+), C-N) circle plus H-1(Omega(-), C-N). We associate with the formal Dirac operator D-A,D-Phi,D-m,D-Gamma delta Sigma an unbounded in L-2(R-n, C-N) operator D-A,D-Phi,D-m,D-B Sigma defined by the Dirac operator D-A,D-Phi,D-m with domain domD(A,Phi,m,B Sigma) subset of H-1(R-n\Sigma, C-N) defined by an interaction conditions. The main aims of the paper are the study of self-adjointmess of the operators D-A,D-Phi,D-m,D-B Sigma for uniformly regular C-2-hypersurfaces Sigma subset of R-n and the essential spectra of D-A,D-Phi,D-m,D-B Sigma for closed C-2-hypersurfaces Sigma subset of R-n.