Singular continuous Floquet operator for periodic quantum systems

AbstractConsider the Floquet operator of a time independent quantum system, acting on a separable Hilbert space, periodically perturbed by a rank one kick: e−iH0Te−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[ and C>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/T02 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−iH0Te−iκT|ϕ〉〈ϕ|)T>0,κ∈R