01345nas a2200145   4500008004100000022001300041245009800054210006900152300001200221490000700233520079700240100002401037700002001061856011801081       2013                        eng d  a1435985500aQuasi-periodic solutions with Sobolev regularity of NLS on Td with a multiplicative potential0 aQuasiperiodic solutions with Sobolev regularity of NLS on Td wit  a229-2860 v153 aWe prove the existence of quasi-periodic solutions for Schrödinger equations with a multiplicative potential on Td , 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 C∞ then the solutions are C∞. 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. © European Mathematical Society 2013.1 aBerti, Massimiliano1 aBolle, Philippe  uhttps://www.math.sissa.it/publication/quasi-periodic-solutions-sobolev-regularity-nls-td-multiplicative-potential