Periodic Dirac operator with dislocation

TL;DR

This paper analyzes the spectral properties of a Dirac operator with a dislocation potential, revealing the behavior of eigenvalues and resonances as functions of the dislocation parameter, with implications for quantum systems with defects.

Contribution

It provides a detailed spectral analysis of a Dirac operator with a dislocation potential, including the behavior of states and construction of specific examples with multiple eigenvalues and resonances.

Findings

States are continuous functions of the dislocation parameter t.

In each gap, states are distinct and can be non-monotone functions of t.

Examples of operators with multiple eigenvalues and resonances in gaps are constructed.

Abstract

We consider a Dirac operator with a dislocation potential on the real line. The dislocation potential is a fixed periodic potential on the negative half-line and the same potential but shifted by real parameter $t$ on the positive half-line. Its spectrum has an absolutely continuous part (the union of bands separated by gaps) plus at most two eigenvalues in each non-empty gap. Its resolvent admits a meromorphic continuation onto a two-sheeted Riemann surface. We prove that it has only two simple poles on each open gap: on the first sheet (an eigenvalue) or on the second sheet (a resonance). These poles are called states and there are no other poles. We prove: 1) each state is a continuous function of $t$, and we obtain its local asymptotic; 2) for each $t$ states in the gap are distinct; 3) in general, a state is non-monotone function of $t$ but it can be monotone for specific potentials; 4) we construct examples of operators, which have: a) one eigenvalue and one resonance in any finite number of gaps; b) two eigenvalues or two resonances in any finite number of gaps; c) two static virtual states in one gap.