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Eigenvalue Bounds for Perturbed Periodic Dirac Operators

Ghada Shuker Jameel, Karl Michael Schmidt

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.

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(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.

Paper Structure

This paper contains 5 sections, 14 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. The periodic equation
  3. Eigenvalue exclusion
  4. Behaviour near the essential spectrum
  5. Asymptotics for large $\mathop{\rm Im}\lambda$

Key Result

Lemma 1

Let $\lambda\in\mathbb C$, $\mathop{\rm Im}\lambda\ge 0$. Then In particular, if $\mathfrak D(\lambda_0) \in [-2, 2]$.

Theorems & Definitions (26)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • Lemma 2
  • proof
  • proof : Proof of Theorem \ref{['thm:compres']}
  • Lemma 3
  • Theorem 3
  • ...and 16 more