Eigenvalue Bounds for Perturbed Periodic Dirac Operators

TL;DR

This paper characterizes the location of eigenvalues of perturbed periodic Dirac operators in the complex plane, showing their proximity to spectral band endpoints or the bands themselves depending on the perturbation's function space.

Contribution

It provides new bounds on eigenvalue locations for perturbed Dirac operators with minimal assumptions on the perturbation.

Findings

Eigenvalues are near spectral band endpoints for small L^1 perturbations.

Eigenvalues are near entire spectral bands for small L^p perturbations, p > 1.

Established relative compactness of certain matrix multiplication operators.

Abstract

We characterise regions in the complex plane that contain all non-embedded eigenvalues of a perturbed periodic Dirac operator on the real line with real-valued periodic potential and a generally non-symmetric matrix-valued perturbation V . We show that the eigenvalues are located close to the end-points of the spectral bands for small V in L^1(R)^{2x2} , but only close to the spectral bands as a whole for small V in L^p(R)^{2x2} , p > 1. As auxiliary results, we prove the relative compactness of matrix multiplication operators in L^{2p}(R)^{2x2} with respect to the periodic operator under minimal hypotheses, and find the asymptotic solution of the Dirac equation on a finite interval for spectral parameters with large imaginary part.