About linearization of infinite-dimensional Hamiltonian systems

TL;DR

This paper proves that under certain conditions, formal symplectic conjugation of infinite-dimensional Hamiltonian systems to a normal form implies actual analytic conjugation, within Gevrey regularity spaces.

Contribution

It establishes conditions under which formal conjugation to a Birkhoff normal form implies actual analytic conjugation in infinite-dimensional Hamiltonian systems.

Findings

Formal conjugation implies analytic conjugation under Diophantine-like conditions.

Results are valid for systems with Gevrey regularity.

Provides a bridge between formal and analytic normal forms in infinite dimensions.

Abstract

This article is concerned with analytic Hamiltonian dynamical systems in infinite dimension in a neighborhood of an elliptic fixed point. Given a quadratic Hamiltonian, we consider the set of its analytic higher order perturbations. We first define the subset of elements which are formally symplectically conjugacted to a (formal) Birkhoff normal form. We prove that if the quadratic Hamiltonian satisfies a Diophantine-like condition and if such a perturbation is formally symplectically conjugated to the quadratic Hamiltonian, then it is also analytically symplectically conjugated to it. Of course what is an analytic symplectic change of variables depends strongly on the choice of the phase space. Here we work on periodic functions with Gevrey regularity.