Eigenvalue bounds for perturbed periodic Dirac 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 $V\in L^1(\mathbb{R}{)}^{2\times 2}$, but only close to the spectral bands as a whole for small $V\in L^p(\mathbb{R}{)}^{2\times 2}$, $p>1$. As auxiliary results, we prove the relative compactness of matrix multiplication operators in $L^{2p}(\mathbb{R}{)}^{2\times 2}$ 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.