×

zbMATH — the first resource for mathematics

Ergodicité et limite semi-classique. (Ergodicity and semi-classical limit). (French) Zbl 0624.58039
Let p(x,y) be a smooth function on phase space \({\mathbb{R}}^{2n}\) and denote by \(h>0\) Planck’s constant. The authors consider the corresponding quantum Hamiltonian \(P(h)=p(x,hD_ x)\) being selfadjoint on \(L^ 2({\mathbb{R}}^ n)\). If the classical flow on a compact energy shell \(\Sigma_{\lambda}=\{(x,y)|\) \(p(x,y)=\lambda \}\) is assumed to be ergodic and if some technical conditions are fulfilled then several interesting results are proved concerning the semiclassical limit of the quantum system. Especially the result is obtained that in the limit \(h\downarrow 0\) almost all (properly normalized) eigenfunctions \(\psi_ h\) of P(h) with eigenvalues \((=energies)\) near \(\lambda\) are distributed according to the Liouville measure \(\sigma_{\lambda}\) on the energy surface \(\Sigma_{\lambda}\), i.e. for any open set \(D\subset {\mathbb{R}}^ n\) it holds that \[ \lim_{h\downarrow 0} \int_{D} | \psi_ h(x)|^ 2 dx = \sigma_{\lambda}(\{(x,y)\in \Sigma_{\lambda}| x\in D\}). \] As a special case the one-dimensional Schrödinger equation is traced. Relations to WKB type results of other authors are discussed.
Reviewer: G.Jetschke

MSC:
58Z05 Applications of global analysis to the sciences
37A99 Ergodic theory
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81Q15 Perturbation theories for operators and differential equations in quantum theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Colin de Verdi?re, Y.: Ergodicit? et fonctions propres du Laplacien, s?minaire Bony-Sj?strand-Meyer 1984-85 no XIII. Commun. Math. Phys.102, 497-502 (1985) · Zbl 0592.58050
[2] Cardelpergher, Nosmas: Propriet?s spectrales d’op?rateurs differentiels asymptotiques autoadjoints. Commun. Partial Differ. Equations9, 137-168 (1984) · Zbl 0586.47033
[3] Duistermaat, J.J.: Oscillatory integrals, Lagrange immersions and unfolding of singularities. Commun. Partial Differ. Equations27, 207-281 (1974) · Zbl 0285.35010
[4] Fedoryuk, Maslov: Semi-classical approximation in quantum mechanics. Amsterdam: Reidel 1981 · Zbl 0458.58001
[5] Helffer, B.: Introduction to the semi-classical analysis for the Schr?dinger operator and applications. Cours ? Nanka? (Chine) (? para?tre)
[6] Comportement semi-classique du spectre des hamiltoniens quantiques elliptiques. Ann. l’Institut FourierXXXI, 169-223 (1981); · Zbl 0451.35022
[7] Calcul fonctionnel par la transformation de Mellin et op?rateurs admissibles. J. Funct. Anal.53, 246-268 (1983); · Zbl 0524.35103
[8] Puits de potentiel g?n?ralis?s et asymptotique semi-classique. Ann. l’Institut Henri Poincar?41, 291-331 (1984)
[9] Multiple wells in the semi-classical limit. I. Commun. Partial Differ. Equations9, 337-408 (1984); · Zbl 0546.35053
[10] Puits multiples en limite semi-classique. II. Interaction mol?culaire, sym?tries, perturbation. Ann. l’Institut Henri Poincar?42, 127-212 (1985)
[11] H?rmander, L.: The Weyl calculus of pseudo-differential operators. Commun. P.A.M.32, 359-443 (1979) · Zbl 0396.47029
[12] Ivrii, V.: A para?tre
[13] Leray, J.: Analyse lagrangienne et m?canique quantique. Coll?ge de France, 1976-1977
[14] Martinez, A.: Estimations de l’effet tunnel pour le double puits. II. Etats hautement excit?s (? para?tre) · Zbl 0666.35069
[15] Petkov, V., Robert, D.: Asymptotique semi-classique du spectre d’hamiltoniens quantiques et trajectoires classiques p?riodiques. Commun. Partial Differ. Equations10, 365-390 (1985) · Zbl 0574.35067
[16] Robert, D.: Autour de l’approximation semi-classique. Cours. de l’Universit? de Nantes et de R?cife (1983). Birkh?user PM 68
[17] Schnirelman, A.: Ergodic properties of eigenfunctions. Usp. Math. Nauk29, 181-182 (1974) · Zbl 0324.58020
[18] Voros, A.: D?veloppements semi-classiques. Th?se Orsay 1977
[19] Zelditch, S.: Eigenfunctions on compact Riemann-surfaces ofg?2. Preprint 1984 (New York)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.