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.