Geodesic
Supersymmetric structures for second order differential operators
Algebra i analiz, Tome 25 (2013) no. 2, pp. 125-154.

Voir la notice de l'article provenant de la source Math-Net.Ru

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.
Mots-clés : eigenvalue splitting, tunnelling effect, Witten–Hodge Laplacian, Kramers–Fokker–Planck operator, Schrödinger operator. 
@article{AA_2013_25_2_a5,
     author = {F. H\'erau and M. Hitrik and J. Sj\"ostrand},
     title = {Supersymmetric structures for second order differential operators},
     journal = {Algebra i analiz},
     pages = {125--154},
     publisher = {mathdoc},
     volume = {25},
     number = {2},
     year = {2013},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/item/AA_2013_25_2_a5/}
}
TY  - JOUR
AU  - F. Hérau
AU  - M. Hitrik
AU  - J. Sjöstrand
TI  - Supersymmetric structures for second order differential operators
JO  - Algebra i analiz
PY  - 2013
SP  - 125
EP  - 154
VL  - 25
IS  - 2
PB  - mathdoc
UR  - https://geodesic-test.mathdoc.fr/item/AA_2013_25_2_a5/
LA  - en
ID  - AA_2013_25_2_a5
ER  - 
%0 Journal Article
%A F. Hérau
%A M. Hitrik
%A J. Sjöstrand
%T Supersymmetric structures for second order differential operators
%J Algebra i analiz
%D 2013
%P 125-154
%V 25
%N 2
%I mathdoc
%U https://geodesic-test.mathdoc.fr/item/AA_2013_25_2_a5/
%G en
%F AA_2013_25_2_a5
F. Hérau; M. Hitrik; J. Sjöstrand. Supersymmetric structures for second order differential operators. Algebra i analiz, Tome 25 (2013) no. 2, pp. 125-154. https://geodesic-test.mathdoc.fr/item/AA_2013_25_2_a5/

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

[2] Bismut J.-M., Lebeau G., The hypoelliptic Laplacian and Ray–Singer metrics, Ann. of Math. Stud., 167, 2008, 367 pp. | MR | Zbl

[3] Bovier A., Eckhoff M., Gayrard V., Klein M., “Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times”, J. Eur. Math. Soc., 6 (2004), 399–424 | DOI | MR | Zbl

[4] Bovier A., Gayrard V., Klein M., “Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues”, J. Eur. Math. Soc., 7 (2005), 69–99 | DOI | MR | Zbl

[5] Dimassi M., Sjöstrand J., Spectral asymptotics in the semi-classical limit, Cambridge Univ. Press, Cambridge, 1999 | MR | Zbl

[6] Eckmann J.-P., Pillet C.-A., Rey-Bellet L., “Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat bath at different temperature”, Comm. Math. Phys., 201 (1999), 657–697 | DOI | MR | Zbl

[7] Franke B., Hwang C.-R., Pai H.-M., Sheu S.-J., “The behavior of the spectral gap under growing drift”, Trans. Amer. Math. Soc., 362 (2010), 1325–1350 | DOI | MR | Zbl

[8] Helffer B., Klein M., Nier F., “Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach”, Mat. Contemp., 26 (2004), 41–85 | MR | Zbl

[9] Helffer B., Sjöstrand J., “Puits multiples en mécanique semi-classique IV, étude du complexe de Witten”, Comm. Partial Differential Equations, 10 (1985), 245–340 | DOI | MR | Zbl

[10] Hérau F., Hitrik M., Sjöstrand J., “Tunnel effect for Fokker–Planck type operators”, Ann. Henri Poincaré, 9:2 (2008), 209–274 | DOI | MR | Zbl

[11] Hérau F., Hitrik M., Sjöstrand J., “Tunnel effect for Kramers–Fokker–Planck type operators: return to equilibrium and applications”, Intern. Math. Res. Notices, 2008:15 (2008), Art. ID rnn057, 48 pp. | MR

[12] Hérau F., Hitrik M., Sjöstrand J., “Tunnel effect and symmetries for Kramers–Fokker–Planck type operators”, J. Inst. Math. Jussieu, 10 (2011), 567–634 | DOI | MR | Zbl

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

[14] Le Peutrec D., “Local WKB construction for Witten Laplacians on manifolds with boundary”, Analysis PDE, 3 (2010), 227–260 | DOI | MR | Zbl

[15] Le Peutrec D., “Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian”, Ann. Fac. Sci. Toulouse Math., 19 (2010), 735–809 | DOI | MR | Zbl

[16] Le Peutrec D., “Small eigenvalues of the Witten Laplacian acting on $p$-forms on a surface”, Asymptot. Anal., 73 (2011), 187–201 | MR | Zbl

[17] Le Peutrec D., Nier F., Viterbo C., “Precise Arrhenius law for $p$-forms: The Witten Laplacian and Morse–Barannikov complex”, Ann. Henri Poincaré, 14:3 (2013), 567–610 | DOI | Zbl

[18] Li Y., Nirenberg L., “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), 85–146 | DOI | MR | Zbl

[19] Ottobre M., Pavliotis G. A., Pravda-Starov K., “Exponential return to equilibrium for hypoelliptic quadratic systems”, J. Funct. Anal., 9 (2012), 4000–4039 | DOI | MR | Zbl

[20] Tailleur J., Tanase-Nicola S., Kurchan J., “Kramers equation and supersymmetry”, J. Statist. Phys., 122 (2006), 557–595 | DOI | MR | Zbl

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