Lower bounds on Anderson-localised eigenfunctions on a strip

TL;DR

This paper investigates the decay rates of eigenfunctions of a random Schrödinger operator on a strip, providing bounds that support the conjecture that eigenfunctions decay at rates between the slowest and fastest Lyapunov exponents.

Contribution

It proves that eigenfunctions decay at rates slower than the fastest Lyapunov exponent and match the slowest along some subsequence, advancing understanding of eigenfunction localization.

Findings

Eigenfunctions decay slower than the fastest Lyapunov exponent.

Existence of subsequences where decay matches the slowest Lyapunov exponent.

Supports the conjecture on decay rate bounds for eigenfunctions.

Abstract

It is known that the eigenfunctions of a random Schr\"odinger operator on a strip decay exponentially, and that the rate of decay is not slower than prescribed by the slowest Lyapunov exponent. A variery of heuristic arguments suggest that no eigenfunction can decay faster than at this rate. We make a step towards this conjecture (in the case when the distribution of the potential is regular enough) by showing that, for each eigenfunction, the rate of exponential decay along any subsequence is strictly slower than the fastest Lyapunov exponent, and that there exists a subsequence along which it is equal to the slowest Lyapunov exponent.