Tunnel effect and symmetries for non-selfadjoint operators
Journées équations aux dérivées partielles (2013), article no. 5, 12 p.

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 $\mathrm{𝒫𝒯}$-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 $\mathrm{𝒫𝒯}$–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.

DOI: 10.5802/jedp.101
Classification: 35P15,  35P20,  47A10,  81Q20,  81Q60,  82C31
Keywords: Non-selfadjoint, supersymmetry, Kramers-Fokker-Planck, tunneling, exponentially small eigenvalue, chain of oscillators, semiclassical limit, $\mathrm{𝒫𝒯}$–symmetry, quadratic operator, Hamilton map
Hitrik, Michael 1

1 Department of Mathematiscs University of California, Los Angeles Los Angeles, CA 90095–1555, United States
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Hitrik, Michael. Tunnel effect and symmetries for non-selfadjoint operators. Journées équations aux dérivées partielles (2013), article  no. 5, 12 p. doi : 10.5802/jedp.101. http://www.numdam.org/articles/10.5802/jedp.101/

[1] J.-M. Bismut, The hypoelliptic Laplacian on the cotangent bundle, J. Amer. Math. Soc. 18 (2005), 379–476. | MR | Zbl

[2] L. Boutet de Monvel, Hypoelliptic operators with double characteristics and related pseudo-differential operators, Comm. Pure Appl. Math. 27 (1974), 585–639. | MR | Zbl

[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 | Zbl

[4] E. Caliceti, S. Graffi, M. Hitrik, and J. Sjöstrand, Quadratic $\mathrm{𝒫𝒯}$–symmetric operators with real spectrum and similarity to self-adjoint operators, J. Phys. A: Math. Theor., 45 (2012), 444007. | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[8] B. Helffer and J. Sjöstrand, Multiple wells in the semiclassical limit. I., Comm. Partial Differential Equations 9 (1984), 337–408. | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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

[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 | Zbl

[14] F. Hérau, M. Hitrik, and J. Sjöstrand, Supersymmetric structures for second order differential operators, St. Petersburg Math. J., to appear. | MR

[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 | Zbl

[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 | Zbl

[17] M. Hitrik and K. Pravda-Starov Spectra and semigroup smoothing for non-elliptic quadratic operators, Math. Ann., 344 (2009), 801–846. | MR | Zbl

[18] H. Risken, The Fokker-Planck equation. Methods of solution and applications, Springer Series in Synergetics, 18 Springer Verlag, Berlin, 1989. | MR | Zbl

[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 | Zbl

[20] B. Simon, Semiclassical analysis of low lying eigenvalues. II. Tunneling, Ann. of Math. 120 (1984), 89–118. | MR | Zbl

[21] J. Sjöstrand, Parametrices for pseudodifferential operators with multiple characteristics, Ark. för Mat. 12 (1974), 85–130. | MR | Zbl

[22] J. Tailleur, S. Tanase-Nicola, J. Kurchan, Kramers equation and supersymmetry, J. Stat. Phys. 122 (2006), 557–595. | MR | Zbl

[23] J. Viola, Spectral projections and resolvent bounds for partially elliptic quadratic differential operators, J. Pseudo-Diff. Op. Appl. 4 (2013), 145–221. | Zbl

[24] E. Witten, Supersymmetry and Morse theory, J. Differential Geom. 17 (1982), 661–692. | MR | Zbl

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