Periodic Dirac operator on the half-line

TL;DR

This paper studies the spectral properties of the Dirac operator with a periodic potential on the half-line, analyzing how states evolve with potential shifts and deriving formulas for potential reconstruction.

Contribution

It introduces a detailed analysis of states and resonances for the Dirac operator with periodic potential, including their dependence on potential shifts and methods for potential recovery.

Findings

States are smooth functions of the shift parameter t.

States are generally non-monotonic functions of t.

Explicit formulas for potential reconstruction are derived.

Abstract

We consider the Dirac operator with a periodic potential on the half-line with the Dirichlet boundary condition at zero. Its spectrum consists of an absolutely continuous part plus at most one eigenvalue in each open gap. The Dirac resolvent admits a meromorphic continuation onto a two-sheeted Riemann surface with a unique simple pole 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. If the potential is shifted by real parameter t, then the continuous spectrum does not change but the states can change theirs positions. We prove that each state is smooth and in general, non-monotonic function of t. We prove that a state is a strictly monotone function of t for a specific potential. Using these results we obtain formulas to recover potentials of special forms.