The Poincare'-Lyapounov-Nekhoroshev theorem

We give a detailed and mainly geometric proof of a theorem by N.N. Nekhoroshev for hamiltonian systems in $n$ degrees of freedom with $k$ constants of motion in involution, where $1 \le k \le n$. This states persistence of $k$-dimensional invariant tori, and local existence of partial action-angle coordinates, under suitable nondegeneracy conditions. Thus it admits as special cases the Poincar\'e-Lyapounov theorem (corresponding to $k=1$) and the Liouville-Arnold one (corresponding to $k = n$), and interpolates between them. The crucial tool for the proof is a generalization of the Poincar\'e map, also introduced by Nekhoroshev.Comment: 21 pages, no figure

Reducibility of 1-d Schroedinger equation with time quasiperiodic unbounded perturbations, I

We study the Schr\"odinger equation on $\R$ with a polynomial potential behaving as $x^{2l}$ at infinity, $1\leq l\in\N$ and with a small time quasiperiodic perturbation. We prove that if the symbol of the perturbation grows at most like $(\xi^2+x^{2l})^{\beta/(2l)}$, with $\beta<l+1$, then the system is reducible. Some extensions including cases with $\beta=2l$ are also proved. The result implies boundedness of Sobolev norms. The proof is based on pseudodifferential calculus and KAM theory

On persistence of invariant tori and a theorem by Nekhoroshev

We give a proof of a theorem by N.N. Nekhoroshev concerning Hamiltonian systems with $n$ degrees of freedom and $s$ integrals of motion in involution, where $1 \le s \le n$. Such a theorem ensures persistence of $s$-dimensional invariant tori under suitable nondegeneracy conditions generalizing Poincar\'e's condition on the Floquet multipliers.Comment: 13 pages, no figure

Exponential times in the one-dimensional Gross--Petaevskii equation with multiple well potential

We consider the Gross-Petaevskii equation in 1 space dimension with a $n$-well trapping potential. We prove, in the semiclassical limit, that the finite dimensional eigenspace associated to the lowest n eigenvalues of the linear operator is slightly deformed by the nonlinear term into an almost invariant manifold M. Precisely, one has that solutions starting on M, or close to it, will remain close to M for times exponentially long with the inverse of the size of the nonlinearity. As heuristically expected the effective equation on M is a perturbation of a discrete nonlinear Schroedinger equation. We deduce that when the size of the nonlinearity is large enough then tunneling among the wells essentially disappears: that is for almost all solutions starting close to M their restriction to each of the wells has norm approximatively constant over the considered time scale. In the particular case of a double well potential we give a more precise result showing persistence or destruction of the beating motions over exponentially long times. The proof is based on canonical perturbation theory; surprisingly enough, due to the Gauge invariance of the system, no non-resonance condition is required

Nekhoroshev theorem for perturbations of the central motion

In this paper we prove a Nekhoroshev type theorem for perturbations of Hamiltonians describing a particle subject to the force due to a central potential. Precisely, we prove that under an explicit condition on the potential, the Hamiltonian of the central motion is quasi-convex. Thus, when it is perturbed, two actions (the modulus of the total angular momentum and the action of the reduced radial system) are approximately conserved for times which are exponentially long with the inverse of the perturbation parameter

Stability of spectral eigenspaces in nonlinear Schrodinger equations

We consider the time-dependent non linear Schrodinger equations with a double well potential in dimensions d =1 and d=2. We prove, in the semiclassical limit, that the finite dimensional eigenspace associated to the lowest two eigenvalues of the linear operator is almost invariant for any time

Almost global existence for a fractional Schrodinger equation on spheres and tori

We study the time of existence of the solutions of the following Schr\"odinger equation i\psi_t = (-\Delta)^s \psi +f(|\psi|^2)\psi, x \in \mathbb S^d, or x\in\T^d where $(-\Delta)^s$ stands for the spectrally defined fractional Laplacian with $s>1/2$ and $f$ a smooth function. We prove an almost global existence result for almost all $s>1/2$