Open AccessQuasi-periodic solutions with Sobolev regularity of NLS on $\mathbb T^d$ with a multiplicative potential
Author(s) -
Massimiliano Berti,
Philippe Bolle
Publication year - 2012
Publication title -
journal of the european mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.549
H-Index - 64
eISSN - 1435-9863
pISSN - 1435-9855
DOI - 10.4171/jems/361
Subject(s) - mathematics , sobolev space , multiplicative function , differentiable function , nonlinear system , measure (data warehouse) , pure mathematics , space (punctuation) , inverse , mathematical analysis , diagonal , linguistics , philosophy , physics , geometry , quantum mechanics , database , computer science
We prove the existence of quasi-periodic solutions for Schrodinger equations with a multiplicative potential on the d-dimensional torus T^d, d ≥ 1, finitely differentiable nonlinearities, and tangential frequencies constrained along a pre-assigned direction. The solutions have only Sobolev regularity both in time and space. If the nonlinearity and the potential are infinitely many times differentiable so are the solutions. The proofs are based on an improved Nash-Moser iterative scheme, which assumes the weakest tame estimates for the inverse linearized operators (“Green functions”) along scales of Sobolev spaces. The key off-diagonal decay estimates of the Green functions are proved via a new multiscale inductive analysis. The main novelty concerns the measure and “complexity” estimates