A new spectral analysis of stationary random Schrodinger operators

Motivated by the long-time transport properties of quantum waves in weakly disordered media, the present work puts random Schrodinger operators into a new spectral perspective. Based on a stationary random version of a Floquet type fibration, we reduce the description of the quantum dynamics to a fibered family of abstract spectral perturbation problems on the underlying probability space. We state a natural resonance conjecture for these fibered operators: in contrast with periodic and quasiperiodic settings, this would entail that Bloch waves do not exist as extended states but rather as resonant modes, and this would justify the expected exponential decay of time correlations. Although this resonance conjecture remains open, we develop new tools for spectral analysis on the probability space, and in particular, we show how ideas from Malliavin calculus lead to rigorous Mourre type results: we obtain an approximate dynamical resonance result and the first spectral proof of the decay of time correlations on the kinetic timescale. This spectral approach suggests a whole new way of circumventing perturbative expansions and renormalization techniques.

Automated summary

Inspired by the transport properties of quantum waves in weakly disordered media, this work introduces a new spectral approach for random Schrodinger operators. A resonance conjecture is proposed for the fibered operators, suggesting that Bloch waves exist as resonant modes rather than extended states. The development of new spectral analysis tools on the probability space leads to rigorous results regarding dynamical resonance and decay of time correlations on the kinetic timescale. This spectral approach offers a fresh perspective on avoiding perturbative expansions and renormalization techniques.