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

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.

[1]  Bernard Helffer,et al.  Puits multiples en mecanique semi-classique iv etude du complexe de witten , 1985 .

[2]  F. Nier.  Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach. , 2004 .

[3]  F. Hérau,et al.  Isotropic Hypoellipticity and Trend to Equilibrium for the Fokker-Planck Equation with a High-Degree Potential , 2004 .

[4]  A. Bovier,et al.  Metastability in Reversible Diffusion Processes I: Sharp Asymptotics for Capacities and Exit Times , 2004 .

[5]  Christiaan C. Stolk,et al.  Semiclassical Analysis for the Kramers–Fokker–Planck Equation , 2004, math/0406275.

[6]  J. Sjoestrand,et al.  Spectra of PT-symmetric operators and perturbation theory , 2004, math-ph/0407052.

[7]  F. Nier,et al.  Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians , 2005 .

[8]  Jean-Michel Bismut,et al.  The hypoelliptic Laplacian on the cotangent bundle , 2005 .

[9]  Kramers Equation and Supersymmetry , 2005, cond-mat/0503545.

[10]  A. Bovier,et al.  Metastability in reversible diffusion processes II. Precise asymptotics for small eigenvalues , 2005 .

[11]  E.Caliceti,et al.  $PT$ symmetric non-selfadjoint operators, diagonalizable and non-diagonalizable, with real discrete spectrum , 2007, 0705.4218.

[12]  J. Sjöstrand,et al.  symmetric non-self-adjoint operators, diagonalizable and non-diagonalizable, with a real discrete spectrum , 2007 .

[13]  F. Hérau,et al.  Tunnel Effect for Kramers–Fokker–Planck Type Operators , 2007, math/0703684.

[14]  F. Hérau,et al.  Tunnel effect for Kramers-Fokker-Planck type operators: return to equilibrium and applications , 2008, 0801.3615.

[15]  D. L. Peutrec.  Small singular values of an extracted matrix of a Witten complex , 2009 .

[16]  Louis Boutet de Monvel,et al.  Hypoelliptic operators with double characteristics and related pseudo-differential operators , 2010 .