  We prove the existence of quasi-periodic solutions for Schroedinger equations
with a multiplicative potential on T^d, d \geq 1, merely 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.
