Approximate normal forms via Floquet-Bloch theory: Nehorosev stability for linear waves in quasiperiodic media

TL;DR

This paper develops an approximate Floquet-Bloch theory to analyze the long-time behavior of linear waves in quasiperiodic media, demonstrating stability and ballistic transport over stretched exponential timescales.

Contribution

It introduces a novel approximate normal form method for Schrödinger operators with quasiperiodic potentials, extending stability results to a broader class of linear waves.

Findings

Establishes Nehoro{}v-type stability for quasiperiodic potentials.

Demonstrates asymptotic ballistic transport up to stretched exponential times.

Extends geometric optics to quasiperiodically perturbed media.

Abstract

We study the long-time behavior of the Schr{\"o}dinger flow in a heterogeneous potential $\lambda$V with small intensity 0<$\lambda$$\ll$1 (or alternatively at high frequencies). The main new ingredient, which we introduce in the general setting of a stationary ergodic potential, is an approximate stationary Floquet--Bloch theory that is used to put the perturbed Schr{\"o}dinger operator into approximate normal form. We apply this approach to quasiperiodic potentials and establish a Nehoro{\v s}ev-type stability result. In particular, this ensures asymptotic ballistic transport up to a stretched exponential timescale exp($\lambda$--1/s) for some s>0. More precisely, the approximate normal form leads to an accurate long-time description of the Schr{\"o}dinger flow as an effective unitary correction of the free flow. The approach is robust and generically applies to linear waves. For classical waves, for instance, this allows to extend diffractive geometric optics to quasiperiodically perturbed media.