We prove the existence of quasi-periodic solutions for Schrödinger equations with a multiplicative potential on T d , d ≥ 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 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.The "separation properties" (i) are quite different for periodic or quasi-periodic solutions. In the first case the singular sites are "separated at infinity", namely the distance between distinct singular sites increases when the Fourier indexes tend to infinity. This property is exploited in [17]. On the contrary, it never holds for quasi-periodic solutions, even for finite dimensional systems. For example, in the ODE case where the small divisors are ω · k, k ∈ Z ν , if the frequency vector ω ∈ R ν is diophantine, then the singular sites k where |ω · k| ≤ ρ are "uniformly distributed" in a neighborhood of the hyperplane ω · k = 0, with nearby indices at distance O(ρ −α ) for some α > 0.This difficulty has been overcome by Bourgain [6], who extended the approach of Craig-Wayne in [17] via a multiscale inductive argument, proving the existence of quasi-periodic solutions of 1-dimensional wave and Schrödinger equations with polynomial nonlinearities. In order to get estimates of the Green functions, Bourgain imposed lower bounds for the determinants of most "singular sub-matrices" along the diagonal. This implies, by a repeated use of the "resolvent identity" (see [24], [10]), a sub-exponentially fast decay of the Green functions. As a consequence, at the end of the iteration, the quasi-periodic solutions are Gevrey regular.At present, KAM theory for 1-dimensional semilinear PDEs has been sufficiently understood, see e.g.[29], [30], [16], but much work remains for PDEs in higher space dimensions, due to the more complex properties of the eigenfunctions and eigenvalues ofThe main difficulties for PDEs in higher dimensions are:1. the multiplicity of the eigenvalues µ j tends to infinity as µ j → +∞, 2. the eigenfunctions ψ j (x) are (in general) "not localized" with respect to the exponentials.Problem 2 has been often bypassed considering pseudo-differential PDEs substituting the multiplicative potential V (x) by a "convolution potential"which, by definition, is diagonal on the exponentials. The scalars m j are called the "Fourier multipliers".Concerning problem 1, since the approach of Craig-Wayne and Bourgain requires only the first order Melnikov non-resonance conditions, it works well, in principle, in case of multiple eigenvalues, in particular for PDEs in higher spatial dimen...

We prove the existence of quasi-periodic solutions for wave equations with a multiplicative potential on T d , d ≥ 1, and finitely differentiable nonlinearities, quasi-periodically forced in time. The only external parameter is the length of the frequency vector. The solutions have Sobolev regularity both in time and space. The proof is based on a Nash-Moser iterative scheme as in [5]. The key tame estimates for the inverse linearized operators are obtained by a multiscale inductive argument, which is more difficult than for NLS due to the dispersion relation of the wave equation. We prove the "separation properties" of the small divisors assuming weaker non-resonance conditions than in [11].

We consider nonisochronous, nearly integrable, a priori unstable Hamiltonian systems with a (trigonometric polynomial) O(μ)-perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with diffusion time T = O((1/μ) ln(1/μ)) by a variational method which does not require the existence of “transition chains of tori” provided by KAM theory. We also prove that our estimate of the diffusion time Td is optimal as a consequence of a general stability result derived from classical perturbation theory

We prove an abstract Nash-Moser implicit function theorem with parameters which covers the applications to the existence of finite dimensional, differentiable, invariant tori of Hamiltonian PDEs with merely differentiable nonlinearities. The main new feature of the abstract iterative scheme is that the linearized operators, in a neighborhood of the expected solution, are invertible, and satisfy the "tame" estimates, only for proper subsets of the parameters. As an application we show the existence of periodic solutions of nonlinear wave equations on Riemannian Zoll manifolds. A point of interest is that, in presence of possibly very large "clusters of small divisors", due to resonance phenomena, it is more natural to expect solutions with only Sobolev regularity.

We prove the existence of small amplitude, (2π/ω)-periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for any frequency ω belonging to a Cantor-like set of asymptotically full measure and for a new set of nonlinearities. The proof relies on a suitable Lyapunov-Schmidt decomposition and a variant of the Nash-Moser implicit function theorem. In spite of the complete resonance of the equation, we show that we can still reduce the problem to a finite dimensional bifurcation equation. Moreover, a new simple approach for the inversion of the linearized operators required by the Nash-Moser scheme is developed. It allows us to deal also with nonlinearities that are not odd and with finite spatial regularity