In this paper we consider Schrödinger equations with sublinear dispersion relation on the one-dimensional torus T:=R/(2πZ). More precisely, we deal with equations of the form ∂tu=iV(ωt)[u] where V(ωt) is a quasi-periodic in time, self-adjoint pseudo-differential operator of the form V(ωt)=V(ωt,x)|D|M+W(ωt), 0<M≤1, |D|:=−∂xx, V is a smooth, quasi-periodic in time function and W is a quasi-periodic time-dependent pseudo-differential operator of order strictly smaller than M. Under suitable assumptions on V and W, we prove that if ω satisfies some non-resonance conditions, the solutions of the Schrödinger equation ∂tu=iV(ωt)[u] grow at most as tη, t→+∞ for any η>0. The proof is based on a reduction to constant coefficients up to smoothing remainders of the vector field iV(ωt) which uses Egorov type theorems and pseudo-differential calculus. The homological equations arising in the reduction procedure involve both time and space derivatives, since the dispersion relation is sublinear. Such equations can be solved by imposing some Melnikov non-resonance conditions on the frequency vector ω.
|Titolo:||Growth of Sobolev norms for time dependent periodic Schrödinger equations with sublinear dispersion|
MONTALTO, RICCARDO (Corresponding)
|Settore Scientifico Disciplinare:||Settore MAT/07 - Fisica Matematica|
Settore MAT/05 - Analisi Matematica
|Data di pubblicazione:||24-ott-2018|
|Digital Object Identifier (DOI):||10.1016/j.jde.2018.10.017|
|Appare nelle tipologie:||01 - Articolo su periodico|