Clustering for the spectrum of \(h\)-pseudodifferential operators with periodic flow on an energy surface
type: Item
instance of: scholarly article
title: Clustering for the spectrum of \(h\)-pseudodifferential operators with periodic flow on an energy surface
zbMATH Open document ID: 0873.58064
author: Sandrine Dozias
published in: Journal of Functional Analysis
publication date: 1997-09-28T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 58J40
Mathematics Subject Classification ID: 32P05
zbMATH DE Number: 1011041
zbMATH Keywords: pseudodifferential operator
zbMATH Keywords: energy surface
zbMATH Keywords: periodic flow
zbMATH Keywords: Hamiltonian flow
zbMATH Keywords: energy band
zbMATH Keywords: spectrum
reviewed by: Peter B. Gilkey
MaRDI profile type: Publication
full work available at URL: jfan.1996.3023
OpenAlex ID: W2074052068
cites work: Q4167627
cites work: Solution of the Schrödinger equation in terms of classical paths
cites work: A semi-classical trace formula for Schrödinger operators
cites work: Formule de Poisson pour les variétés riemanniennes
cites work: Spectre D'Un hamiltonien quantique et mecanique classique
cites work: Q5180060
cites work: Sur le spectre des opérateurs elliptiques à bicaracteristiques toutes périodiques
cites work: Q4321661
cites work: Q4305698
cites work: The spectrum of positive elliptic operators and periodic bicharacteristics
cites work: Q4139272
cites work: Calcul fonctionnel par la transformation de Mellin et opérateurs admissibles
cites work: Q3679624
cites work: Q3736112
cites work: Semiclassical principal symbols and Gutzwiller's trace formula
cites work: Autour de l'approximation semi-classique. (Around semiclassical approximation)
cites work: On the volume of manifolds all of whose geodesics are closed
cites work: Asymptotics of eigenvalue clusters for the Laplacian plus a potential
DOI: 10.1006/JFAN.1996.3023
description: scientific article; zbMATH DE number 1011041