On quantum Floquet theorem

Abstract

We consider the Schrödinger equation $ih{\partial }_t\ddot{I}ˆ=H\ddot{I}ˆ$, $\ddot{I}ˆ=\ddot{I}ˆ(\cdot ,t)\in L^2(\mathbb{T})$. The operator $H=-{\partial }_x^2+V(x,t)$ includes smooth potential $V$, which is assumed to be time $T$-periodic. Let $W=W(t)$ be the fundamental solution of this linear ODE system on $L^2(\mathbb{T})$. Then according to terminology from Lyapunov-Floquet theory, $\mathcal{M}=W(T)$ is the monodromy operator. We prove that $\mathcal{M}$ is unitarily conjugated to $\exp (-\frac{T}{ih}{\partial }_x^2)+\mathcal{C}$, where $\mathcal{C}$ is a compact operator with an arbitrarily small norm.