Abstract :
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.
