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Spectrum of the perturbed Landau-Dirac operator

Vincent Bruneau, Pablo Miranda

TL;DR

This work analyzes the spectrum of the 2D Dirac operator in a constant magnetic field under compactly supported, non-definite-sign perturbations. By reducing to Toeplitz-type operators via a Foldy–Wouthuysen transform and employing spectral-flow concepts, the authors derive a three-term asymptotic expansion for the eigenvalue counting near each Landau–Dirac level, with the leading terms governed by logarithmic capacities of the perturbation supports. They introduce weights and an effective operator framework for smooth perturbations, revealing how the interaction of the perturbation components $V_1$, $V_2$, and the magnetic field shape the accumulation (finite vs infinite) of eigenvalues and even yield level-specific behavior. An index-theoretic appendix extends Birman–Schwinger-type methods to non-semibounded, matrix-valued settings and establishes an encirclement property that helps manage negative perturbations. Overall, the paper extends previous definite-sign results to the non-definite setting and highlights the pivotal role of capacity geometry and spectral-flow techniques in magnetic Dirac perturbations.

Abstract

In this article, we consider the Dirac operator with constant magnetic field in $\mathbb R^2$. Its spectrum consists of eigenvalues of infinite multiplicities, known as the Landau-Dirac levels. Under compactly supported perturbations, we study the distribution of the discrete eigenvalues near each Landau-Dirac level. Similarly to the Landau (Schrödinger) operator, we demonstrate that a three-terms asymptotic formula holds for the eigenvalue counting function. One of the main novelties of this work is the treatment of some perturbations of variable sign. In this context we explore some remarkable phenomena related to the finiteness or infiniteness of the discrete eigenvalues, which depend on the interplay of the different terms in the matrix perturbation.

Spectrum of the perturbed Landau-Dirac operator

TL;DR

This work analyzes the spectrum of the 2D Dirac operator in a constant magnetic field under compactly supported, non-definite-sign perturbations. By reducing to Toeplitz-type operators via a Foldy–Wouthuysen transform and employing spectral-flow concepts, the authors derive a three-term asymptotic expansion for the eigenvalue counting near each Landau–Dirac level, with the leading terms governed by logarithmic capacities of the perturbation supports. They introduce weights and an effective operator framework for smooth perturbations, revealing how the interaction of the perturbation components , , and the magnetic field shape the accumulation (finite vs infinite) of eigenvalues and even yield level-specific behavior. An index-theoretic appendix extends Birman–Schwinger-type methods to non-semibounded, matrix-valued settings and establishes an encirclement property that helps manage negative perturbations. Overall, the paper extends previous definite-sign results to the non-definite setting and highlights the pivotal role of capacity geometry and spectral-flow techniques in magnetic Dirac perturbations.

Abstract

In this article, we consider the Dirac operator with constant magnetic field in . Its spectrum consists of eigenvalues of infinite multiplicities, known as the Landau-Dirac levels. Under compactly supported perturbations, we study the distribution of the discrete eigenvalues near each Landau-Dirac level. Similarly to the Landau (Schrödinger) operator, we demonstrate that a three-terms asymptotic formula holds for the eigenvalue counting function. One of the main novelties of this work is the treatment of some perturbations of variable sign. In this context we explore some remarkable phenomena related to the finiteness or infiniteness of the discrete eigenvalues, which depend on the interplay of the different terms in the matrix perturbation.
Paper Structure (16 sections, 16 theorems, 126 equations)

This paper contains 16 sections, 16 theorems, 126 equations.

Table of Contents

  1. The $2D$ Magnetic Landau-Dirac operators
  2. Results
  3. Logarithmic capacity
  4. Weights for open sets
  5. Main results
  6. Effective operator for smooth perturbations
  7. Proof of the main results
  8. Reduction to Toeplitz operators
  9. Eigenvalue distribution for the Toeplitz operators ${\mathcal{P}}_{q} {{\mathcal{V}}} {\mathcal{P}}_{q}$
  10. Proof of Proposition \ref{['thm2']}
  11. Proof of Theorem \ref{['thm1']} and Theorem \ref{['theo1_b']}
  12. Reduction to Toeplitz operators $p_0 v p_0$ for smooth potentials
  13. Appendix: Proof of Proposition \ref{['Diag']} and the encirclement property
  14. The index of a pair of projections
  15. Proof of Proposition \ref{['Diag']}
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let ${{\mathcal{V}}} \in L^{\infty}({\mathbb R}^2,{\mathbb R}^2 )$ be a function given by defV essentially supported in ${\mathcal{K}}$, a compact set with Lipschitz boundary. Suppose that there exist $\Omega$ an open bounded set in ${\mathbb R}^2$, $w$ a weight in $\Omega$ and constants $C_2, C_3\g Then, for any $q \in {\mathbb Z}$, the eigenvalue counting function satisfies where $\ln_2(\lambda

Theorems & Definitions (35)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Theorem 2.5
  • Corollary 2.6
  • Corollary 2.7
  • Remark 2.8
  • Proposition 3.1
  • Lemma 3.2
  • ...and 25 more