Singular continuous Floquet operator for periodic quantum systems

Consider the Floquet operator of a time independent quantum system, acting on a separable Hilbert space, periodically perturbed by a rank one kick: e −iH0T e −iκT |φ φ| where T is the period, κ the coupling constant, and H0 is a pure point self-adjoint operator, bounded from below. Under some hypotheses on the vector φ, cyclic w.r.t. H0 we prove the following: • If the gaps between the eigenvalues (λn) are such that λn+1 − λn Cn −γ for some γ ∈ ]0, 1[ andC >0, then the Floquet operator of the perturbed system is purely singular continuous T -a.e. • If H0 is the Hamiltonian of the one-dimensional rotator on L2(R/T0Z) and the ratio 2πT/T 2 0 is irrational, then the Floquet operator is purely singular continuous as soon as κT = 0 (2π). We also establish an integral formula for the family (e −iH0T e −iκT |φ φ| )T>0, κ∈R.  2004 Elsevier Inc. All rights reserved.