Clustering for the spectrum of \(h\)-pseudodifferential operators with periodic flow on an energy surface
Let \(P(h)\) be an \(h\)-pseudo-differential operator with \(h\)-regular symbol \(p(h)\). The author studies the repartition of the eigenvalues of \(P\) on a small neighbourhood of an energy where the Hamiltonian flow associated with the principal symbol \(h\) is periodic. The author uses a geometric lemma and perturbation techniques to obtain results. The paper is divided into 3 sections: (1) Introduction, (2) Approximation by a Hamiltonian whose flow is periodic in a band: (2.1) Construction of a Hamiltonian with periodic flow in an energy band, (2.2) Comparison of the spectrum, (3) Proofs of theorems, (3.1) Distribution of the eigenvalues, (3.2) Counting function relative to each interval, (3.3) The generalized distribution theorem, (3.4) Non-vanishing sub-principal symbol, (3.5) Concluding remarks. The author shows that the eigenvalues of an \(h\)-pseudo-differential operator are localized in intervals of width \(O(h^2)\) centered on a sequence of points which are computed in terms of \(h\) and the classical characteristics of the periodic trajectories on the energy surface associated with the energy level \(E\), their period \(T\), their action \(S\), and their Maslov index. The author also studies the number of eigenvalues in each of these small intervals.
scientific article; zbMATH DE number 1011041