Two-Dimensional Dirac Operators with Interactions on Unbounded Smooth Curves

Abstract: We consider the 2D Dirac operator with singular potentials (Formula presented.) where (Formula presented.) σ_j,i=1,2.3 are Pauli matrices, a=(a,_1,a2 is the magnetic potential with a_j ε L^∞(R²),Φ ∈ L^∞(R) is the electrostatic potential, Q_sin = QδΓ is the singular potential with the strength matrix Q = (Q_ij)² _i,j=1, and δ_Γ is the delta-function with support on a C²curve Γ, which is the common boundary of the domains Ω± R². We associate with the formal Dirac operator Da,Φ,Q_sin an unbounded operator D_A,Φ,Q in L²(R², C²) generated by D_a,Φ with a domain in H¹(Ω₊,C²)⊕H¹(Ω₋,C²) consisting of functions satisfying interaction conditions on Γ We study the self-adjointness of the operator D_A,Φ,Q and its essential spectrum for potentials and curves Γ 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 Ω_±.