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We prove the existence of quasi-periodic solutions for Schroedinger equationswith a multiplicative potential on T^d, d \geq 1, merely differentiablenonlinearities, and tangential frequencies constrained along a pre-assigneddirection. The solutions have only Sobolev regularity both in time and space.If the nonlinearity and the potential are infinitely many times differentiableso are the solutions. The proofs are based on an improved Nash-Moser iterativescheme, which assumes the weakest tame estimates for the inverse linearizedoperators ("Green functions") along scales of Sobolev spaces. The keyoff-diagonal decay estimates of the Green functions are proved via a newmultiscale inductive analysis. The main novelty concerns the measure and"complexity" estimates.

Quasi-periodic solutions with Sobolev regularity of NLS on T^d with a multiplicative potential