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Eigenvalue Asymptotics near a flat band in presence of a slowly decaying potential

Pablo Miranda, Daniel Parra

Abstract

We provide eigenvalue asymptotics for a Dirac-type operator on $\mathbb Z^n$, $n\geq 2$, perturbed by multiplication operators that decay as $|μ|^{-γ}$ with $γ<n$. We show that the eigenvalues accumulate near the value of the flat band at a ''semiclassical'' rate with a constant that encodes the structure of the flat band. Similarly, we show that this behaviour can be obtained also for a Laplace operator on a periodic graph.

Eigenvalue Asymptotics near a flat band in presence of a slowly decaying potential

Abstract

We provide eigenvalue asymptotics for a Dirac-type operator on , , perturbed by multiplication operators that decay as with . We show that the eigenvalues accumulate near the value of the flat band at a ''semiclassical'' rate with a constant that encodes the structure of the flat band. Similarly, we show that this behaviour can be obtained also for a Laplace operator on a periodic graph.

Paper Structure

This paper contains 13 sections, 10 theorems, 93 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Spectral Theory for a Dirac operator on Zn
  3. Discrete Dirac operator
  4. Integral decomposition
  5. Spectrum and resolvent of H0
  6. Perturbed operator and Main Result
  7. Proof
  8. Some notation and results on compact operators
  9. Effective Hamiltomian
  10. Eigenvalue counting function for the effective Hamiltonian
  11. The Laplacian on a particular Z2-periodic graph
  12. A simple example of a Z2-periodic graph with a flat band
  13. Admissible perturbations and eigenvalue asymptotics.

Key Result

Proposition 2.1

The operator $H_0$ satisfy that where $h_0$ denotes the multiplication operator by the real analytic function on $L^2(\mathbb{T}^n,\mathbb{C}^{n+1})$ given by

Figures (2)

  • Figure 1: Two views of the three band functions for $n=2$. The negative band and the flat band only touch at $(0,0)$.
  • Figure :

Theorems & Definitions (20)

  • Proposition 2.1: Pa17
  • Definition 3.1
  • Theorem 3.2
  • Remark 3.3
  • Proposition 4.1: Cwikel-Birman-Solomyak
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Proposition 4.4
  • ...and 10 more