Growth of Sobolev norms for time dependent periodic Schrödinger equations with sublinear dispersion
Articolo
Data di Pubblicazione:
2018
Citazione:
Growth of Sobolev norms for time dependent periodic Schrödinger equations with sublinear dispersion / R. Montalto. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - (2018 Oct 24). [Epub ahead of print] [10.1016/j.jde.2018.10.017]
Abstract:
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), 00. 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.
Tipologia IRIS:
01 - Articolo su periodico
Elenco autori:
R. Montalto
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