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

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