Perturbation of an Eigen-value from a Dense Point Spectrum : an Example

We study a perturbed Floquet Hamiltonian K + βV depending on a coupling constant β. The spectrum σ(K) is assumed to be pure point and dense. We pick up an eigen-value, namely 0 ∈ σ(K), and show the existence of a function λ(β) defined on I ⊂ R such that λ(β) ∈ σ(K + βV ) for all β ∈ I, 0 is a point of density for the set I, and the Rayleigh-Schrödinger perturbation series represents an asymptotic series for the function λ(β). All ideas are developed and demonstrated when treating an explicit example but some of them are expected to have an essentially wider range of application. Key-Words: Floquet Hamiltonians, dense point spectrum, perturbation theory February 1997 CPT-97/P.3454 anonymous ftp or gopher: cpt.univ-mrs.fr ∗Unité Propre de Recherche 7061 and PhyMaT, Université de Toulon et du Var, BP 132, F-83957 La Garde cedex, France [email protected] Department of Mathematics, Faculty of Nuclear Science, CTU, Trojanova 13, CZ-120 00 Prague, Czech Republic [email protected] [email protected] CONTENTS 1