Non-selfadjoint perturbations of selfadjoint operators in 2 dimensions I

This is the first in a series of works devoted to small non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators 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 is $\gg h$ (or sometimes only $\gg h^2$) and bounded from above by $h^{\delta }$ for some $\delta >0$. We get a complete asymptotic description of all eigenvalues in certain rectangles $[-1\mathrm{/}C,1\mathrm{/}C]+iϵ[F_0-1\mathrm{/}C,F_0+1\mathrm{/}C]$.