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St. Petersburg Mathematical Journal

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Supersymmetric structures for second order differential operators


Authors: F. Hérau, M. Hitrik and J. Sjöstrand
Original publication: Algebra i Analiz, tom 25 (2013), nomer 2.
Journal: St. Petersburg Math. J. 25 (2014), 241-263
MSC (2010): Primary 81Q20, 81Q60, 82C22, 82C31; Secondary 35P15, 47A75, 47B44
DOI: https://doi.org/10.1090/S1061-0022-2014-01288-5
Published electronically: March 12, 2014
MathSciNet review: 3114853
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Abstract: Necessary and sufficient conditions are obtained for a real semiclassical partial differential operator of order two to possess a supersymmetric structure. For the operator coming from a chain of oscillators coupled to two heat baths, it is shown that no smooth supersymmetric structure can exist for a suitable interaction potential, provided that the temperatures of the baths are different.


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  • 1. J.-M. Bismut, The hypoelliptic Laplacian on the cotangent bundle, J. Amer. Math. Soc. 18 (2005), no. 2, 379-476. MR 2137981 (2006f:35036)
  • 2. J.-M. Bismut and G. Lebeau, The hypoelliptic Laplacian and Ray-Singer metrics, Ann. of Math. Stud. 167 (2008), 367 p. MR 2441523 (2010b:58042)
  • 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), no. 4, 399-424. MR 2094397 (2006b:82112)
  • 4. A. Bovier, V. Gayrard, and M. Klein, Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues, J. Eur. Math. Soc. 7 (2005), no. 1, 69-99. MR 2120991 (2006b:82113)
  • 5. M. Dimassi and J. Sjöstrand, Spectral asymptotics in the semi-classical limit, London Math. Society Lecture Notes Ser., 268, Cambridge Univ. Press, Cambridge, 1999. MR 1735654 (2001b:35237)
  • 6. J.-P. Eckmann, C.-A. Pillet, and L. Rey-Bellet, Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures, Comm. Math. Phys. 201 (1999), no. 3, 657-697. MR 1685893 (2000d:82025)
  • 7. B. Franke, C.-R. Hwang, H.-M. Pai, and S.-J. Sheu, The behavior of the spectral gap under growing drift, Trans. Amer. Math. Soc. 362 (2010), no. 3, 1325-1350. MR 2563731 (2010m:35343)
  • 8. 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 (2005i:58025)
  • 9. B. Helffer and J. Sjöstrand, Puits multiples en mécanique semi-classique IV, Étude du complexe de Witten, Comm. Partial Differential Equations 10 (1985), no. 3, 245-340. MR 780068 (87i:35162)
  • 10. F. Hérau, M. Hitrik, and J. Sjöstrand, Tunnel effect for Fokker-Planck type operators, Ann. Henri Poincaré 9 (2008), no. 2, 209-274. MR 2399189 (2009k:35214)
  • 11. -, Tunnel effect for Kramers-Fokker-Planck type operators: Return to equilibrium and applications, Int. Math. Res. Not. IMRN 2008, no. 15, 48 pp. MR 2438070 (2009j:35045)
  • 12. -, Tunnel effect and symmetries for Kramers-Fokker-Planck type operators, J. Inst. Math. Jussieu 10 (2011), no. 3, 567-634. MR 2806463 (2012h:35249)
  • 13. M. Hitrik and K. Pravda-Starov, Spectra and semigroup smoothing for non-elliptic quadratic operators, Math. Ann. 344 (2009), no. 4, 801-846. MR 2507625 (2011g:35278)
  • 14. D. Le Peutrec, Local WKB construction for Witten Laplacians on manifolds with boundary, Anal. PDE 3 (2010), no. 3, 227-260. MR 2672794 (2011j:58035)
  • 15. -, Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian, Ann. Fac. Sci. Toulouse Math.(6) 19 (2010), no. 3-4, 735-809. MR 2790817 (2012c:58042)
  • 16. -, Small eigenvalues of the Witten Laplacian acting on $ p$-forms on a surface, Asymptot. Anal. 73 (2011), no. 4, 187-201. MR 2859124 (2012i:58025)
  • 17. D. Le Peutrec, F. Nier, and C. Viterbo, Precise Arrhenius law for $ p$-forms: The Witten Laplacian and Morse-Barannikov complex, Ann. Henri Poincaré, 2012. MR 3035640
  • 18. Y. Li, L. Nirenberg, The distance function to the boundary, Finsler's geometry and the singular set of viscosity solutions of some Hamilton-Jacobi equations, Comm. Pure Appl. Math. 58 (2005), no. 1, 85-146. MR 2094267 (2005k:35042)
  • 19. M. Ottobre, G. A. Pavliotis, and K. Pravda-Starov, Exponential return to equilibrium for hypoelliptic quadratic systems, J. Funct. Anal. 262 (2012), no. 9, 4000-4039. MR 2899986
  • 20. J. Tailleur, S. Tanase-Nicola, and J. Kurchan, Kramers equation and supersymmetry, J. Statist. Phys. 122 (2006), no. 4, 557-595. MR 2213943 (2007h:81125)
  • 21. E. Witten, Supersymmetry and Morse theory, J. Differential Geom. 17 (1982), no. 4, 661-692. MR 683171 (84b:58111)

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Additional Information

F. Hérau
Affiliation: Laboratoire de Mathématiques Jean Leray, Université de Nantes, 2, rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France — and — UMR 6629 CNRS
Email: herau@univ-nantes.fr

M. Hitrik
Affiliation: Department of Mathematics, University of California, Los Angeles, CA 90095-1555
Email: hitrik@math.ucla.edu

J. Sjöstrand
Affiliation: IMB, Université de Bourgogne, 9, Av. A. Savary, BP 47870, FR-21078 Dijon Cédex, France — and — UMR 5584 CNRS
Email: johannes.sjostrand@u-bourgogne.fr

DOI: https://doi.org/10.1090/S1061-0022-2014-01288-5
Keywords: Eigenvalue splitting, tunnelling effect, Witten--Hodge Laplacian, Kramers--Fokker--Planck operator, Schr\"odinger operator
Received by editor(s): October 25, 2012
Published electronically: March 12, 2014
Additional Notes: Supported by the French Agence Nationale de la Recherche, NOSEVOL project, ANR 2011 BS01019 01
Dedicated: In memory of Vladimir Buslaev
Article copyright: © Copyright 2014 American Mathematical Society

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