Low lying eigenvalues of randomly curved quantum waveguides

TL;DR

This paper derives explicit bounds on the lowest eigenvalues of randomly curved quantum waveguides, providing insights into the effects of weak disorder on spectral properties relevant to Anderson localization.

Contribution

It introduces a method to estimate the probability of low eigenvalues in randomly curved waveguides under weak disorder, advancing understanding of spectral behavior in such systems.

Findings

Explicit lower bounds on first eigenvalues in weak disorder regime

Probability estimates for low lying eigenvalues

Relevance to Anderson localization phenomena

Abstract

We consider the negative Dirichlet Laplacian on an infinite waveguide embedded in $\RR^2$, and finite segments thereof. The waveguide is a perturbation of a periodic strip in terms of a sequence of independent identically distributed random variables which influence the curvature. We derive explicit lower bounds on the first eigenvalue of finite segments of the randomly curved waveguide in the small coupling (i.e. weak disorder) regime. This allows us to estimate the probability of low lying eigenvalues, a tool which is relevant in the context of Anderson localization for random Schr\"odinger operators.