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On the measure of spectra for discrete Schrödinger operators on periodic graphs

Natalia Saburova

Abstract

We consider discrete Schrödinger operators $H_{μQ}=Δ+μQ$ with real periodic potentials $Q$ on periodic graphs, where $Δ$ is the adjacency operator and $μ\in\mathbb R$ is a coupling constant. The spectra of the operators consist of a finite number of closed intervals (bands). In the large coupling regime, we obtain an asymptotic upper bound for the measure of the spectrum of $H_{μQ}$ which depends essentially on a "degeneracy degree" of the potential $Q$. This result extends the result of Y. Last obtained for the one-dimensional lattice $\mathbb Z$ to the case of general periodic graphs. It also may serve as a certain quantitative complement to the recent criterion of J. Fillman for the measure of the spectrum of $H_{μQ}$ to go to zero as $μ\to\infty$.

On the measure of spectra for discrete Schrödinger operators on periodic graphs

Abstract

We consider discrete Schrödinger operators with real periodic potentials on periodic graphs, where is the adjacency operator and is a coupling constant. The spectra of the operators consist of a finite number of closed intervals (bands). In the large coupling regime, we obtain an asymptotic upper bound for the measure of the spectrum of which depends essentially on a "degeneracy degree" of the potential . This result extends the result of Y. Last obtained for the one-dimensional lattice to the case of general periodic graphs. It also may serve as a certain quantitative complement to the recent criterion of J. Fillman for the measure of the spectrum of to go to zero as .

Paper Structure

This paper contains 9 sections, 5 theorems, 57 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Periodic graphs and edge indices.
  4. Cycles and their indices
  5. Spectra of the Schrödinger operators on periodic graphs.
  6. Proof of the main result
  7. Minimum spanning trees
  8. A choice of the periodic graph embedding
  9. Upper bounds on the spectral bands widths

Key Result

Theorem 1.1

F26 Assume $Q:{\mathcal{V}}\to{\mathbb R}$ is a periodic potential on a ${\mathbb Z}^d$-periodic graph ${\mathcal{G}}=({\mathcal{V}},{\mathcal{E}})$. Then if and only if ${\mathcal{G}}$ does not contain a path with infinitely many distinct vertices such that the potential is constant along this path. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: a) A ${\mathbb Z}^2$-periodic graph ${\mathcal{G}}\subset{\mathbb R}^2$; $\mathfrak{e}_1,\mathfrak{e}_2$ are the standard basis of ${\mathbb R}^2$; the unit cell $\Omega=[0,1)^2$ is shaded; b) the quotient graph ${\mathcal{G}}_*={\mathcal{G}}/{\mathbb Z}^2$ is a graph on the 2-dimensional torus ${\mathbb R}^2/{\mathbb Z}^2$; the torus is obtained by identification of opposite sides of $\Omega$.

Theorems & Definitions (5)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 3.1
  • Proposition 3.3