Perturbations of selfadjoint operators with periodic classical flow

We consider non-selfadjoint perturbations of a self-adjoint $h$-pseudodifferential operator in dimension 2. In the present work we treat the case when the classical flow of the unperturbed part is periodic and the strength $ϵ$ of the perturbation satisfies $h^{\delta_0} <\epsilon \le \epsilon_0$forsome$\delta_0\in ]0,1/2[$andasufficientlysmall$\epsilon _0>0$. We get a complete asymptotic description of all eigenvalues in certain rectangles $[-1\mathrm{/}C,1\mathrm{/}C]+iϵ[F_0-1\mathrm{/}C,F_0+1\mathrm{/}C]$. In particular we are able to treat the case when $ϵ>0$ is small but independent of $h$.