Useful bounds on the extreme eigenvalues and vectors of matrices for Harper's operators

TL;DR

This paper introduces three novel methods to bound the extreme eigenvalues and eigenvectors of matrices arising from random walks on groups, with applications to Schrödinger operators and harmonic oscillators.

Contribution

It develops three distinct approaches—geometric, analytical limit, and probabilistic—to effectively bound eigenvalues of matrices related to group-based random walks, surpassing traditional bounds.

Findings

The geometric approach uses eigenspace geometry and uncertainty principles.

The limit approach relates matrices to the harmonic oscillator eigenstructure.

The probabilistic method employs hitting times in Markov chains.

Abstract

In analyzing a simple random walk on the Heisenberg group we encounter the problem of bounding the extreme eigenvalues of an $n\times n$ matrix of the form $M=C+D$ where $C$ is a circulant and $D$ a diagonal matrix. The discrete Schr\"odinger operators are an interesting special case. The Weyl and Horn bounds are not useful here. This paper develops three different approaches to getting good bounds. The first uses the geometry of the eigenspaces of $C$ and $D$, applying a discrete version of the uncertainty principle. The second shows that, in a useful limit, the matrix $M$ tends to the harmonic oscillator on $L^2(\mathbb{R})$ and the known eigenstructure can be transferred back. The third approach is purely probabilistic, extending $M$ to an absorbing Markov chain and using hitting time arguments to bound the Dirichlet eigenvalues. The approaches allow generalization to other walks on other groups.