Absence of point spectrum for unitary operators

AbstractLet us consider the time-dependent Schrödinger equation,iφt=−Δφ+V(x,t)φ, on the Hilbert space L2(Rn), where V(x,t) is a repulsive periodic time-dependent potential, with period T. We denote by (U(t,s))(t,s)∈R×R its associated propagator. First, using a multiplier method, we rule out the existence of regular eigenvectors of the Floquet operator U(T,0). Secondly, strengthening the hypotheses on the potential V, we prove that the spectrum of U(T,0) does not contain any eigenvalues, by means of positive commutator methods