Quasi-Töplitz Functions in KAM Theorem

Abstract

We define and describe the class of Quasi-Töplitz functions. We then prove an abstract KAM theorem where the perturbation is in this class. We apply this theorem to a Non-Linear-Schrödinger equation on the torus T d , thus proving existence and stability of quasi-periodic solutions.

Reducibility of quantum harmonic oscillator on Rd with differential and quasi-periodic in time potential

“…We remark that KAM theory is almost well-developed for nonlinear Hamiltonian PDEs in 1-d context. See [4,26,30,31,32,33,34,35,36,37,44,51] [17,18,23,25,27,45] for n-d results. See [8] for an almost complete picture of recent KAM theory.…”

A KAM Result on Compact Lie Groups

“…A very natural question is whether the reducibility results by Eliasson-Kuksin [14,15] (at least in the simplified case considered in Procesi-Xu [26]) can be extended also to this setting. In other words, this would mean to be able to extend the result of the present paper to the case of arbitrary rank.…”

“…In order to show that the set of parameters has positive measure they need to study carefully the asymptotics of the eigenvalues (the so called Töplitz-Lipschitz condition). We mention also the papers [18,26,25] which make use of the conservation of momentum in order to fully diagonalise the matrix.…”

KAM Theorem for a Hamiltonian System with Sublinear Growth Frequencies at Infinity

“…We refer the reader to [3,4,5,6,9,11,12,15,16,17,19,21,22,23,25,26,28] for more information. Nevertheless, there are few results on the Kolmogorov-Arnold-Moser (KAM) theorem for the Hamiltonian (1.1).…”