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

Abstract

We study a perturbed Floquet Hamiltonian $K+\beta V$ depending on a coupling constant $\beta$. The spectrum $\sigma (K)$ is assumed to be pure point and dense. We pick up an eigen-value, namely $0\in \sigma (K)$, and show the existence of a function $\lambda (\beta )$ defined on $I\subset \mathbb{R}$ such that $\lambda (\beta )\in \sigma (K+\beta V)$ for all $\beta \in I$, 0 is a point of density for the set $I$, and the Rayleigh-Schrödinger perturbation series represents an asymptotic series for the function $\lambda (\beta )$. 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

Cited by in corpus (2)

• The characteristic function for Jacobi matrices with applications
• Perturbation of an eigenvalue from a dense point spectrum: a general Floquet Hamiltonian