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

Duclos, P.; Stovicek, P.; Vittot, M.

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

Authors

Duclos, P.

Stovicek, P.

Vittot, M.

Description

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 $0\inσ(K)$, and show the existence of a function $λ(β)$ defined on $I\subset\R$ such that $λ(β) \in σ(K+βV)$ for all $β\in I$, 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.
Latex, 24 pages, 51 K