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

In the present paper, we focus on the reducibility of an almost-periodic linear Hamiltonian system

$$\frac{dX}{dt}=J[A+\epsilon Q(t)]X,X\in {\mathbb{R}}^{2d},$$

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 $\epsilon$ is a parameter which is sufficiently small. Using some non-resonant and non-degeneracy conditions, rapidly convergent methods prove that, for most sufficiently small $\epsilon$, 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.