A new spectral analysis of stationary random Schrödinger operators

Mitia Duerinckx, Christopher Shirley

Published 2020-04-25Version 1

Motivated by the long-time transport properties of quantum waves in weakly disordered media, and more precisely by the quantum diffusion conjecture, the present work puts random Schr\"odinger operators into a new spectral perspective, based on a stationary random version of a Floquet type fibration. The latter reduces the description of the quantum dynamics to a fibered family of abstract spectral perturbation problems on the underlying probability space, and we state a resonance conjecture that would justify part of the expected diffusive behavior. Although this resonance conjecture remains open, we develop new tools for the spectral analysis on the probability space, and in particular we show in the Gaussian setting how ideas from Malliavin calculus lead to partial Mourre type results, which in turn yield the first spectral proof of the exponential decay of time correlations. This spectral approach suggests a whole new way of circumventing perturbative expansions and renormalization techniques.