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Spectral Gaps for Jacobi Matrices on Graphs

Jonathan Breuer, Eyal Seelig

Abstract

We study bounds on eigenvalue gaps for finite quotients of periodic Jacobi matrices on trees. We prove an Alon-Boppana type bound for the spectral gap and a comparison result for other eigenvalue gaps.

Spectral Gaps for Jacobi Matrices on Graphs

Abstract

We study bounds on eigenvalue gaps for finite quotients of periodic Jacobi matrices on trees. We prove an Alon-Boppana type bound for the spectral gap and a comparison result for other eigenvalue gaps.
Paper Structure (5 sections, 10 theorems, 75 equations)

This paper contains 5 sections, 10 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. An Alon-Boppana Bound for Lifts of Jacobi Matrices
  3. Comparison Bounds
  4. Appendix
  5. Proof of Theorem \ref{['thm-greenberg-jacobi']}

Key Result

Theorem 1.1

Let $J_0$ be a Jacobi matrix on a finite graph $\mathcal{G}_0$. Let $\mathcal{G}_n$ be a sequence of finite covers of $\mathcal{G}_0$ such that $|V(\mathcal{G}_n)| \to \infty$, and let $J_n$ be the lift of $J_0$ to $\mathcal{G}_n$. Let $J_T$ be the lift of $J_0$ to the universal cover $T$ and let $\

Theorems & Definitions (25)

  • Theorem 1.1
  • Conjecture 1.1
  • Theorem 1.2
  • Remark
  • Remark
  • Corollary 1.3: Bound on the spectral gap
  • proof
  • Remark
  • Lemma 2.1
  • proof
  • ...and 15 more