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.