  We consider a perturbed Floquet Hamiltonian $-i\partial_t + H + \beta
V(\omega t)$ in the Hilbert space $L^2([0,T],E,dt)$. Here $H$ is a self-adjoint
operator in $E$ with a discrete spectrum obeying a growing gap condition,
$V(t)$ is a symmetric bounded operator in $E$ depending on $t$
$2\pi$-periodically, $\omega = 2\pi/T$ is a frequency and $\beta$ is a coupling
constant. The spectrum $Spec(-i\partial_t + H)$ of the unperturbed part is pure
point and dense in $R$ for almost every $\omega$. This fact excludes
application of the regular perturbation theory. Nevertheless we show, for
almost all $\omega$ 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 Rayleigh-Schrodinger
series are asymptotic to the perturbed eigen-value and the perturbed
eigen-vector.
