Perturbation of an Eigen-value from a Dense Point Spectrum: a General Floquet Hamiltonian

We consider a perturbed Floquet Hamiltonian −i∂t + H + βV (ωt) in the Hilbert space L([0, T ],H, dt). Here H is a self-adjoint operator in H with a discrete spectrum obeying a growing gap condition, V (t) is a symmetric bounded operator in H depending on t 2π-periodically, ω = 2π/T is a frequency and β is a coupling constant. The spectrum Spec(−i∂t + H) of the unperturbed part is pure point and dense in R for almost every ω. This fact excludes application of the regular perturbation theory. Nevertheless we show, for almost all ω and provided V (t) is sufficiently smooth, that the perturbation theory still makes sense, however, with two modifications. First, the coupling constant is restricted to a set I which need not be an interval but 0 is still a point of density of I. Second, the RayleighSchrodinger series are asymptotic to the perturbed eigen-value and the perturbed eigen-vector. CPT-97/P.3559 November 18, 1997 ftp://cpt.univ-mrs.fr http://www.cpt.univ-mrs.fr [email protected] [email protected] [email protected] Typeset by AMS-TEX 1 2 P. DUCLOS, P. ŠŤOVÍČEK AND M. VITTOT