Non-selfadjoint perturbations of selfadjoint operators in 2 dimensions II. Vanishing Averages

This is the second in a series of works devoted to small non-selfadjoint perturbations of selfadjoint semiclassical pseudodifferential operators in dimension 2. As in our previous work, we consider the case when the classical flow of the unperturbed part is periodic. Under the assumption that the flow average of the leading perturbation vanishes identically, we show how to obtain a complete asymptotic description of the individual eigenvalues in certain domains in the complex plane, provided that the strength of the perturbation $\epsilon$ is $\gg h^{1/2}$, or sometimes only $\gg h$, and enjoys the upper bound $\epsilon={\cal O}(h^{\delta})$, for some $\delta>0$.