Tail bounds for the Dyson series of random Schr\"odinger equations

TL;DR

This paper establishes tail bounds for the Dyson series in random Schr"odinger equations, providing insights into eigenfunction localization and delocalization at specific time scales without complex harmonic analysis.

Contribution

It introduces an elementary method using noncommutative Khintchine inequality to derive tail bounds for Dyson series terms in random Schr"odinger equations.

Findings

Tail bounds effective at time scales ~λ^{-2+ε}

Estimates on eigenfunction localization and delocalization

Applicability to Floquet states in time-periodic potentials

Abstract

We study Schr\"odinger equations on $\mathbb{Z}^d$ and $\mathbb{R}^d$, $d\geq 2$ with random potentials of strength $\lambda$. Our main result gives tail bounds for the terms of the Dyson series that are effective at time scales on the order of $\lambda^{-2+\varepsilon}$. As corollaries, we obtain estimates on the frequency localization and spatial delocalization of approximate eigenfunctions in the spirit of works by Schlag-Shubin-Wolff and T. Chen. These estimates also apply to Floquet states associated to time-periodic potentials. Our proof is elementary in that we use neither sophisticated harmonic analysis nor diagrammatic arguments. Instead, we use only the noncommutative Khintchine inequality from random matrix theory combined with pointwise dispersive estimates for the free Schr\"odinger equation.