On Self-Adjointness of 3D—Dirac Operators with Singular Potentials

Abstract

Abstract We consider the Dirac operator on ℝ3, D A , Φ , Q s = ∑ j = 1 3 α j ( i ∂ x j + A j ( x ) ) + α 0 m − Φ ( x ) I 4 + Q s , with magnetic potential A (x) = (A 1(x), A 2(x), A 3(x))) and electrostatic potential Φ(x), where αj,j = 0, 1, 2, 3 are the Dirac matrices, Qs = ΓδS is a singular potential where δS is the Dirac δ—function with support on an enough smooth surface S ⊂ ℝ3 divided ℝ3 on two open domains Ω+, Ω− with common unbounded boundary S, and Γ is 4 × 4 matrix. We associate with the formal Dirac operator D A , Φ, Qs the unbounded in the Hilbert space L 2(ℝ3, ℂ4) operator D with domain defined by some interaction conditions on the surface S. The purpose of the paper is to give conditions of the self-adjointness of the operator D.