Two-Dimensional Dirac Operators with Interactions on Unbounded Smooth Curves

We consider the 2D Dirac operator with singular potentials D-A,D-Phi,D-Qsin u(x) = (D-A,D-Phi + Q(sin)) u(x), x is an element of R-2,R- (1) where D-a,D-Phi = Sigma(2)(j=1)sigma(j) (i partial derivative(xj) + a(j)) + sigma(3)m + Phi I-2; (2)here sigma(j), j = 1, 2, 3, are Pauli matrices, a = (a(1), a(2)) is the magnetic potential with a(j) is an element of L-infinity (R-2), Phi is an element of L-infinity(R) is the electrostatic potential, Q(sin) = Q delta(Gamma) is the singular potential with the strength matrix Q = (Q(ij))(i,j=1)(2), and delta(Gamma) is the delta-function with support on a C-2- curve Gamma, which is the common boundary of the domains Omega(+/-) subset of R-2. We associate with the formal Dirac operator D-alpha,D- Phi,D-Qsin an unbounded operator D-A,D- Phi,D-Q in L-2 (R-2, C-2) generated by D-a,D- Phi with a domain in H-1(Omega(+), C-2) circle plus H-1 (Omega(-), C-2) consisting of functions satisfying interaction conditions on Gamma. We study the self-adjointness of the operator D-A,D- Phi,D-Q and its essential spectrum for potentials and curves Gamma slowly oscillating at infinity. We also study the splitting of the interaction problems into two boundary problems describing the confinement of particles in the domains Omega +/-.