We prove the existence of quasi-periodic solutions for Schrödinger equations with a multiplicative potential on , 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 then the solutions are . 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.
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Massimiliano Berti, Philippe Bolle, Quasi-periodic solutions with Sobolev regularity of NLS on with a multiplicative potential. J. Eur. Math. Soc. 15 (2013), no. 1, pp. 229–286DOI 10.4171/JEMS/361