A new spectral analysis of stationary random Schrödinger operators

Abstract

Motivated by the long-time transport properties of quantum waves in weakly disordered media, the present work puts random Schrödinger 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.

A remark on a surprising result by Bourgain in homogenization

“…In fact, while the classical corrector theory is optimal in view of the strong effective approximation of the solution operator in L 2 (R d × Ω), the results described here beg for the development of a novel higher-order corrector theory in a weak sense in probability. This shares some close connection with results in [8], and the investigation of Conjecture 1.3 in this spirit is postponed to a future work.…”

Cherenkov radiation with massive bosons and quantum friction

“…In the present contribution, we propose a new construction that aims to cure regularity issues and bring us back to the realm of regular Mourre theory. As inspired by our previous work [14], the crucial point is that we modify the construction of natural conjugates in a form that is no longer that of second quantization. To show the wide applicability of this modification procedure, we illustrate our results on two paradigmatic systems:…”

“…As is well known, e.g. [11,Section 1.3] and [14,Lemma 5.6], random Schrödinger operators can be viewed as particular instances of Pauli-Fierz models. This comparison was actually our original motivation for the present contribution, in link with our previous work [14].…”

“…[11,Section 1.3] and [14,Lemma 5.6], random Schrödinger operators can be viewed as particular instances of Pauli-Fierz models. This comparison was actually our original motivation for the present contribution, in link with our previous work [14]. More precisely, given a stochastically translation-invariant Gaussian field V on R d , constructed on a probability space Ω, a Gaussian chaos decomposition of L 2 (Ω) ensures that the random Schrödinger operator…”

“…In other words, the action of the Gaussian field is viewed as the interaction with a bosonic field. Via this isomorphism, the fiber decomposition (1.5) for H g is precisely equivalent to the Floquet-Bloch type fibration that we introduced in [14],…”