The optimal spectral gap for regular and disordered harmonic networks of oscillators

TL;DR

This paper determines the precise dependence of the spectral gap on system size for harmonic oscillator networks in various dimensions, using a novel eigenvalue analysis method.

Contribution

It introduces a new approach to analyze the spectral gap of harmonic networks, providing the first sharp N dependence results under different physical conditions.

Findings

Sharp N dependence of spectral gap established

Applicable to various physical assumptions and dimensions

New eigenvalue analysis method developed

Abstract

We consider one-dimensional chains and multi-dimensional networks of harmonic oscillators coupled to two Langevin heat reservoirs at different temperatures. Each particle interacts with its nearest neighbors by harmonic potentials and all individual particles are confined by harmonic potentials, too. In this article, we provide, for the first time, the sharp N dependence of the spectral gap of the associated generator under various physical assumptions and for different spatial dimensions. Our method of proof relies on a new approach to analyze a non self-adjoint eigenvalue problem involving low-rank non-hermitian perturbations of auxiliary discrete Schr\"odinger operators.