  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 $I\subset\R$ 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-Schr\"odinger 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.
