Spectra for Semiclassical Operators with Periodic Bicharacteristics in Dimension Two

Abstract

We study the distribution of eigenvalues for selfadjoint h-pseudodifferential operators in dimension two, arising as perturbations of selfadjoint operators with a periodic classical flow. When the strength ε of the perturbation is ≪ h, the spectrum displays a cluster structure, and assuming that ε ≫ h 2 (or sometimes ≫ h N 0 , for N 0 > 1 large), we obtain a complete asymptotic description of the individual eigenvalues inside subclusters, corresponding to the regular values of the leading symbol of the perturbation, averaged along the flow.

Concentration and Non-Concentration for the Schrödinger Evolution on Zoll Manifolds

“…We note that, thanks to the semiclassical Weyl's law [12], one knows that s 0 ( ) > 0. In the case where d = 2 and N ≪ ǫ ≪ √ (where N is some positive exponent related to the clustering of the unperturbed operator), a much stronger result was for instance obtained in [26]. In fact, it was proved in this reference that, near regular values of I g (V ), one can obtain an asymptotic expansion of the eigenvalues with a level spacing which is exactly of order ǫ 2 .…”

“…The study of the spectral properties of the operator − 1 2 ∆ g + V in this geometric context is a problem which has a long history in microlocal analysis starting with the works of Duistermaat-Guillemin [14,23,24], Weinstein [51] and Colin de Verdière [9]. Many other important results on the fine structure of the spectrum of Zoll manifolds were obtained both in the microlocal framework [25,47,48,53,54], and in the semiclassical setting [8,28,26] -see also [30,31] in the nonselfadjoint setting.…”

“…One often makes this assumption when studying the spectral properties of semiclassical operators with periodic bicharacteritics (see [41,26] for instance) in order to to keep the nice cluster structure of the the spectrum of the operator −∆ g , see [9].…”

Asymptotic lattices, good labellings, and the rotation number for quantum integrable systems

“…See Figure 2. If J is proper (and Ĵ has no subprincipal symbol), the result follows from the cluster structure of the spectrum of pseudodifferential operators with periodic characteristics, see for instance [27,24,14,31]. In our case, we do not impose the properness of J and there is no restriction on subprincipal symbols; the result is still valid because we restrict the joint spectrum to a small B (it would not hold for the usual spectrum of Ĵ alone).…”

Asymptotic Lattices, Good Labellings, and the Rotation Number for Quantum Integrable Systems

“…See Figure 2. If J is proper (and Ĵ has no subprincipal symbol), the result follows from the cluster structure of the spectrum of pseudodifferential operators with periodic characteristics, see for instance [18,16,10,22]. In our case, we do not impose the properness of J and there is no restriction on subprincipal symbols; the result is still valid because we restrict the joint spectrum to a small B (it would not hold for the usual spectrum of Ĵ alone).…”