369 research outputs found
The Poincare'-Lyapounov-Nekhoroshev theorem
We give a detailed and mainly geometric proof of a theorem by N.N.
Nekhoroshev for hamiltonian systems in degrees of freedom with
constants of motion in involution, where . This states
persistence of -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
) and the Liouville-Arnold one (corresponding to ), 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 with a polynomial potential
behaving as at infinity, and with a small time
quasiperiodic perturbation. We prove that if the symbol of the perturbation
grows at most like , with , then the
system is reducible. Some extensions including cases with 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 degrees of freedom and integrals of motion in involution,
where . Such a theorem ensures persistence of -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
-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
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 stands for the spectrally
defined fractional Laplacian with and a smooth function. We prove
an almost global existence result for almost all
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
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