Clustering for the Spectrum ofh-pseudodifferential Operators with Periodic Flow on an Energy Surface

We study the repartition of the eigenvalues of anh-pseudodifferential operator in a small neighbourhood of an energy where the Hamiltonian flow associated with the principal symbol of the operator is periodic. The proof is based on a geometric lemma and perturbation techniques to obtain the results from the known theorem in the case of a periodic flow in an energy band (B. Helffer and D. Robert,Ann. Inst. H. Poincaré41(1984), 291–331).