Approximate normal forms via Floquet-Bloch theory. Part 1: Nehoro\v{s}ev stability for linear waves in quasiperiodic media

Mitia Duerinckx (1) , Antoine Gloria (2, 1) , Christopher Shirley (2)

1 ULB - Département de Mathématique [Bruxelles]
2 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions

We study the long-time behavior of the Schrödinger flow in a heterogeneous potential λV with small intensity 0 < λ ≪ 1. 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. In this first contribution we apply this approach to quasiperiodic potentials and establish a Nehorošev-type stability result. In particular, this ensures as-ymptotic ballistic transport up to a stretched exponential timescale exp(λ − 1 s) for some s > 0. More precisely, the approximate normal form decomposition leads to long-time effective equations obtained by adding suitable unitary corrections to the free flow. The approach is robust and generically applies to linear waves. For classical waves, for instance , this allows to revisit diffractive geometric optics in quasiperiodically perturbed media. The application to the random setting is postponed to a companion article.

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Théorie spectrale [math.SP]

Mitia Duerinckx, Antoine Gloria, Christopher Shirley. Approximate normal forms via Floquet-Bloch theory. Part 1: Nehoro\v{s}ev stability for linear waves in quasiperiodic media. 2018. ⟨hal-01876846v1⟩