Centre de diffusion de revues académiques mathématiques

 
 
 
 

Journées équations aux dérivées partielles

Table des matières de ce volume | Article précédent | Article suivant
Michael Hitrik
Tunnel effect and symmetries for non-selfadjoint operators
Journées équations aux dérivées partielles (2013), Exp. No. 5, 12 p., doi: 10.5802/jedp.101
Article PDF
Class. Math.: 35P15, 35P20, 47A10, 81Q20, 81Q60, 82C31
Mots clés: Non-selfadjoint, supersymmetry, Kramers-Fokker-Planck, tunneling, exponentially small eigenvalue, chain of oscillators, semiclassical limit, $\mathcal{PT}$–symmetry, quadratic operator, Hamilton map

Résumé - Abstract

We study low lying eigenvalues for non-selfadjoint semiclassical differential operators, where symmetries play an important role. In the case of the Kramers-Fokker-Planck operator, we show how the presence of certain supersymmetric and $\mathcal{PT}$-symmetric structures leads to precise results concerning the reality and the size of the exponentially small eigenvalues in the semiclassical (here the low temperature) limit. This analysis also applies sometimes to chains of oscillators coupled to two heat baths, but when the temperatures of the baths are different, we show that the supersymmetric approach may break down. We also discuss $\mathcal{PT}$–symmetric quadratic differential operators with real spectrum and characterize those that are similar to selfadjoint operators. This talk is based on joint works with Emanuela Caliceti, Sandro Graffi, Frédéric Hérau, and Johannes Sjöstrand.

Bibliographie

[1] J.-M. Bismut, The hypoelliptic Laplacian on the cotangent bundle, J. Amer. Math. Soc. 18 (2005), 379–476.  MR 2137981 |  Zbl 1065.35098
[2] L. Boutet de Monvel, Hypoelliptic operators with double characteristics and related pseudo-differential operators, Comm. Pure Appl. Math. 27 (1974), 585–639.  MR 370271 |  Zbl 0294.35020
[3] A. Bovier, M. Eckhoff, V. Gayrard, and M. Klein, Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times, J. Eur. Math. Soc. 6 (2004), 399–424.  MR 2094397 |  Zbl 1076.82045
[4] E. Caliceti, S. Graffi, M. Hitrik, and J. Sjöstrand, Quadratic $\mathcal{PT}$–symmetric operators with real spectrum and similarity to self-adjoint operators, J. Phys. A: Math. Theor., 45 (2012), 444007.  MR 2991874 |  Zbl 1263.81190
[5] J.-P. Eckmann, C.-A. Pillet, L. Rey-Bellet, Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat bath at different temperature, Comm. Math. Phys. 201 (1999), 657–697.  MR 1685893 |  Zbl 0932.60103
[6] B. Helffer, M. Klein, and F. Nier, Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach, Mat. Contemp. 26 (2004), 41–85.  MR 2111815 |  Zbl 1079.58025
[7] B. Helffer and F. Nier, Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians, Lecture Notes in Mathematics, 1862. Springer-Verlag, Berlin, 2005.  MR 2130405 |  Zbl 1072.35006
[8] B. Helffer and J. Sjöstrand, Multiple wells in the semiclassical limit. I., Comm. Partial Differential Equations 9 (1984), 337–408.  MR 740094 |  Zbl 0546.35053
[9] B. Helffer and J. Sjöstrand, Puits multiples en limite semi-classique. II. Interaction moléculaire. Symétries. Perturbation, Ann. Inst. H. Poincaré Phys. Th. 42 (1985), 127–212. Numdam |  MR 798695 |  Zbl 0595.35031
[10] B. Helffer and J. Sjöstrand, Multiple wells in the semiclassical limit. III. Interaction through non-resonant wells, Math. Nachr. 124 (1985), 263–313.  MR 827902 |  Zbl 0597.35023
[11] B. Helffer and J. Sjöstrand, Puits multiples en mécanique semi-classique. IV. Etude du complexe de Witten, Comm. Partial Differential Equations, 10 (1985), 245–340.  MR 780068 |  Zbl 0597.35024
[12] F. Hérau, M. Hitrik, and J. Sjöstrand, Tunnel effect for Kramers-Fokker-Planck type operators: return to equilibrium and applications, International Math Res Notices, 15 (2008), Article ID rnn057, 48pp.  Zbl 1151.35012
[13] F. Hérau, M. Hitrik, and J. Sjöstrand, Tunnel effect and symmetries for Kramers-Fokker-Planck type operators, J. Inst. Math. Jussieu 10 (2011), 567–634.  MR 2806463 |  Zbl 1223.35246
[14] F. Hérau, M. Hitrik, and J. Sjöstrand, Supersymmetric structures for second order differential operators, St. Petersburg Math. J., to appear.  MR 3114853
[15] F. Hérau and F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal. 171 (2004), 151–218.  MR 2034753 |  Zbl 1139.82323
[16] F. Hérau, J. Sjöstrand, and C. Stolk, Semiclassical analysis for the Kramers-Fokker-Planck equation, Comm. Partial Differential Equations 30 (2005), 689–760.  MR 2153513 |  Zbl 1083.35149
[17] M. Hitrik and K. Pravda-Starov Spectra and semigroup smoothing for non-elliptic quadratic operators, Math. Ann., 344 (2009), 801–846.  MR 2507625 |  Zbl 1171.47038
[18] H. Risken, The Fokker-Planck equation. Methods of solution and applications, Springer Series in Synergetics, 18 Springer Verlag, Berlin, 1989.  MR 987631 |  Zbl 0665.60084
[19] B. Simon, Semiclassical analysis of low lying eigenvalues. I. Nondegenerate minima: asymptotic expansions, Ann. Inst. H. Poincaré Sect. A. (N.S.) 38 (1983), 295–308. Numdam |  MR 708966 |  Zbl 0526.35027
[20] B. Simon, Semiclassical analysis of low lying eigenvalues. II. Tunneling, Ann. of Math. 120 (1984), 89–118.  MR 750717 |  Zbl 0626.35070
[21] J. Sjöstrand, Parametrices for pseudodifferential operators with multiple characteristics, Ark. för Mat. 12 (1974), 85–130.  MR 352749 |  Zbl 0317.35076
[22] J. Tailleur, S. Tanase-Nicola, J. Kurchan, Kramers equation and supersymmetry, J. Stat. Phys. 122 (2006), 557–595.  MR 2213943 |  Zbl 1149.81013
[23] J. Viola, Spectral projections and resolvent bounds for partially elliptic quadratic differential operators, J. Pseudo-Diff. Op. Appl. 4 (2013), 145–221.  Zbl 1285.47049
[24] E. Witten, Supersymmetry and Morse theory, J. Differential Geom. 17 (1982), 661–692.  MR 683171 |  Zbl 0499.53056
Copyright Cellule MathDoc 2018 | Crédit | Plan du site