On quantum Floquet theorem

We consider the Schrödinger 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 a 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 the terminology from Lyapunov–Floquet theory, $\mathcal M=W(T)$ is the monodromy operator. We prove that $\mathcal M$ is unitarily conjugated to $D+\mathcal C$, where $D$ is diagonal in the standard Fourier basis, while $\mathcal C$ is a compact operator with an arbitrarily small norm.