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 $\delta$-shell interaction of strength $\tau \in \mathbb{R}\setminus \{-2,0,2\}$ supported on a broken line of opening angle $2\omega$ with $\omega \in (0,\frac{\pi }{2})$. 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 $\tau <0$, also on the strength of the interaction, but does not depend on $\omega$. As the main result, we prove that for any $N\in \mathbb{N}$ and strength $\tau \in (-\infty ,0)\setminus \{-2\}$ 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 $\omega$ is sufficiently small. Moreover, we obtain an explicit estimate on $\omega$ sufficient for this property to hold. For $\tau \in (0,\infty )\setminus \{2\}$, the discrete spectrum consists of at most one simple eigenvalue.

28 pages, 3 figures