On quantum Floquet theorem

Abstract

We consider the Schr\"odinger equation $ih\partial_t\psi = H\psi$, $\psi=\psi(\cdot,t)\in L^2({\mathbb T})$. The operator $H = -\partial^2_x + V(x,t)$ 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, ${\cal M}=W(T)$ is the monodromy operator. We prove that ${\cal M}$ is unitarily conjugated to $\exp\big(-\frac{T}{ih} \partial^2_x\big) + {\cal C}$, where ${\cal C}$ is a compact operator with an arbitrarily small norm.