Quasi-periodic solutions with Sobolev regularity of NLS on ${\mathbb{T}}^d$ with a multiplicative potential

Abstract

We prove the existence of quasi-periodic solutions for Schrödinger equations with a multiplicative potential on ${\mathbb{T}}^d,d\ge 1$, 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 $C^{\infty }$ then the solutions are $C^{\infty }$. 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.