Eigenvalue Bounds for Perturbed Periodic Dirac Operators

2024/10/16 by Ghada Shuker Jameel, Jameel, Ghada Shuker, Karl Michael Schmidt +1
Computer Science · Mathematics · #34L40 #47A55 #47B28 #81Q15 #Advanced Mathematical Modeling in Engineering #Differential Equations and Numerical Methods #FOS: Mathematics #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2410.12467

openalex publication_date 2024/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 L¹(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.

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