Approximate normal forms via Floquet-Bloch theory: Nehorosev stability for linear waves in quasiperiodic media
Résumé
We study the long-time behavior of the Schrödinger flow in a heterogeneous potential λV with small intensity 0<λ≪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 Schrödinger operator into approximate normal form. We apply this approach to quasiperiodic potentials and establish a Nehorošev-type stability result. In particular, this ensures asymptotic ballistic transport up to a stretched exponential timescale exp(λ−1/s) for some s>0. More precisely, the approximate normal form leads to an accurate long-time description of the Schrödinger 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.
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