Low lying spectrum of weak-disorder quantum waveguides

TL;DR

This paper investigates the low-energy spectrum of a quantum waveguide with weak random disorder, providing bounds on the lowest eigenvalue and establishing spectral localization through multiscale analysis.

Contribution

It introduces new bounds on the lowest eigenvalue for weak-disorder waveguides and applies multiscale analysis to prove spectral localization.

Findings

Deterministic bounds on the lowest eigenvalue

Probabilistic bounds on eigenvalue position

Spectral localization established

Abstract

We study the low-lying spectrum of the Dirichlet Laplace operator on a randomly wiggled strip. More precisely, our results are formulated in terms of the eigenvalues of finite segment approximations of the infinite waveguide. Under appropriate weak-disorder assumptions we obtain deterministic and probabilistic bounds on the position of the lowest eigenvalue. A Combes-Thomas argument allows us to obtain so-called 'initial length scale decay estimates' at they are used in the proof of spectral localization using the multiscale analysis.