Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-28T18:27:48.767Z Has data issue: false hasContentIssue false
  • English
  • Français

Tunnel effect and symmetries for Kramers–Fokker–Planck type operators

Part of: Spectral theory and eigenvalue problems

Published online by Cambridge University Press:  12 May 2011

Frédéric Hérau ,
Michael Hitrik  and
Johannes Sjöstrand
Frédéric Hérau
Affiliation:
Laboratoire de Mathématiques, Jean Leray, Université de Nantes, 2 rue de la Houssinière, BP 92208, 44322 Nantes cedex 3, France (herau@univ-nantes.fr) and UMR 6629, CNRS, France
Michael Hitrik
Affiliation:
Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA (hitrik@math.ucla.edu)
Johannes Sjöstrand
Affiliation:
IMB, Université de Bourgogne, 9, Avenue A. Savary, BP 47870, 21078 Dijon Cédex, France and UMR 5584, CNRS, France (johannes.sjostrand@u-bourgogne.fr)
Get access Rights & Permissions [Opens in a new window]

Abstract

We study operators of Kramers–Fokker–Planck type in the semiclassical limit, assuming that the exponent of the associated Maxwellian is a Morse function with a finite number n0 of local minima. Under suitable additional assumptions, we show that the first n0 eigenvalues are real and exponentially small, and establish the complete semiclassical asymptotics for these eigenvalues.

Résumé

Nous étudions des opérateurs de type Kramers–Fokker–Planck dans la limite semi-classique quand l'exposant du maxwellien associé est une fonction de Morse avec un nombre fini n0 de minima locaux. Sous des hypothèses supplémentaires convenables, nous montrons que les premières n0 valeurs propres sont réelles et exponentiellement petites et nous établissons leur asymptotique semi-classique complète.

Keywords

semiclassical analysisFokker–Plancktunnel effecteigenvalue splitting

MSC classification

Secondary: 35P15: Estimation of eigenvalues, upper and lower bounds 35P20: Asymptotic distribution of eigenvalues and eigenfunctions
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 10 , Special Issue 3: A Special Issue in Honour of Louis Boutet de Monvel and Pierre Schapira , July 2011 , pp. 567 - 634
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bismut, J. M., The hypoelliptic Laplacian on the cotangent bundle, J. Am. Math. Soc. 18 (2005), 379–476.CrossRefGoogle Scholar
2.de Monvel, L. Boutet, Hypoelliptic operators with double characteristics and related pseudo-differential operators, Commun. Pure Appl. Math. 27 (1974), 585–639.CrossRefGoogle Scholar
3.Bovier, A., Eckhoff, M., Gayrard, V. and Klein, M., Metastability in reversible diffusion processes, I, Sharp asymptotics for capacities and exit times, J. Eur. Math. Soc. 6(4) (2004), 399–424.Google Scholar
4.Bovier, A., Gayrard, V. and Klein, M., Metastability in reversible diffusion processes, II, Precise asymptotics for small eigenvalues, J. Eur. Math. Soc. 7(1) (2005), 69–99.Google Scholar
5.Graffi, S., Calicetti, E. and Sjöstrand, J., Spectra of PT-symmetric operators and perturbation theory, J. Phys. A38(1) (2005), 185–193.Google Scholar
6.Graffi, S., Calicetti, E. and Sjöstrand, J., PT symmetric non-self-adjoint operators, diagonalizable and non-diagonalizable, with real discrete spectrum, J. Phys. A40(33) (2007), 10 155–10 170.Google Scholar
7.Helffer, B. and Nier, F., Hypoelliptic estimates and spectral theory for Fokker–Planck operators and Witten Laplacians, Lecture Notes in Mathematics, Volume 1862 (Springer, 2005).CrossRefGoogle Scholar
8.Helffer, B. and Sjöstrand, J., Puits multiples en mécanique semi-classique, IV, Étude du complexe de Witten, Commun. PDEs 10(3) (1985), 245–340.CrossRefGoogle Scholar
9.Helffer, B., Klein, M. and Nier, F., Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach, Mat. Contemp. 26 (2004), 41–85.Google Scholar
10.Hérau, F. and Nier, F., Isotropic hypoellipticity and trend to equilibrium for the Fokker–Planck equation with a high-degree potential, Arch. Ration. Mech. Analysis 171(2) (2004), 151–218.CrossRefGoogle Scholar
11.Hérau, F., Sjöstrand, J. and Stolk, C., Semiclassical analysis for the Kramers–Fokker–Planck equation, Commun. PDEs 30(5–6) (2005), 689–760.CrossRefGoogle Scholar
12.Hérau, F., Hitrik, M. and Sjöstrand, J., Tunnel effect for Fokker–Planck type operators, Annales Inst. H. Poincaré 9(2) (2008), 209–274.CrossRefGoogle Scholar
13.Hérau, F., Hitrik, M. and Sjöstrand, J., Tunnel effect for Kramers–Fokker–Planck type operators: return to equilibrium and applications, Int. Math. Res. Not. 2008 (2008), rnn057.Google Scholar
14.Le Peutrec, D., Small singular values of an extracted matrix of a Witten complex, CUBO 11(4) (2009), 49–57.Google Scholar
15.Lebeau, G., Le bismutien, Séminaire Équations aux Dérivées Partielles, Volume I.1–15 (École Polytechnique, 2004–2005).Google Scholar
16.Nier, F., Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach, Journées Équations aux Dérivées Partielles, Exposé VIII (École Polytechnique, 2004).Google Scholar
17.Tailleur, J., Tanase-Nicola, S. and Kurchan, J., Kramers equation and supersymmetry, J. Statist. Phys. 122(4) (2006), 557–595.CrossRefGoogle Scholar