Abstract

Let us consider the time-dependent Schrödinger equation,i φt = - Δ φ + 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. © 2007 Elsevier Inc. All rights reserved.