On the reducibility of a class of almost-periodic linear Hamiltonian systems and its application in Schrödinger equation

Abstract

In the present paper, we focus on the reducibility of an almost-periodic linear Hamiltonian system $ \frac{dX}{dt} = J[A+\varepsilon Q(t)]X, X\in \mathbb{R}^{2d} , $ where $ J $ is an anti-symmetric symplectic matrix, $ A $ is a symmetric matrix, $ Q(t) $ is an analytic almost-periodic matrix with respect to $ t $, and $ \varepsilon $ is a parameter which is sufficiently small. Using some non-resonant and non-degeneracy conditions, rapidly convergent methods prove that, for most sufficiently small $ \varepsilon $, the Hamiltonian system is reducible to a constant coefficients Hamiltonian system through an almost-periodic symplectic transformation with similar frequencies as $ Q(t) $. At the end, an application to Schrödinger equation is given.

Keywords

• reducibility
• almost-periodic
• kam method
• hamiltonian system