Spectral analysis of the Dirac operator with a singular interaction on a broken line

Abstract

We consider the one-parametric family of self-adjoint realizations of the two-dimensional massive Dirac operator with a Lorentz scalar δ-shell interaction of strength τ∈R\{−2,0,2} supported on a broken line of opening angle 2ω with ω∈(0,π2). The essential spectrum of any such self-adjoint realization is symmetric with respect to the origin with a gap around zero whose size depends on the mass and, for τ < 0, also on the strength of the interaction, but does not depend on ω. As the main result, we prove that for any N∈N and strength τ ∈ (−∞, 0)\{−2} the discrete spectrum of any such self-adjoint realization has at least N discrete eigenvalues, with multiplicities taken into account, in the gap of the essential spectrum provided that ω is sufficiently small. Moreover, we obtain an explicit estimate on ω sufficient for this property to hold. For τ ∈ (0, ∞)\{2}, the discrete spectrum consists of at most one simple eigenvalue.