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Jean-François Bony; Frédéric Hérau; Laurent Michel
Tunnel effect for semiclassical random walk
Journées équations aux dérivées partielles (2014), Exp. No. 6, 18 p., doi: 10.5802/jedp.109
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Résumé - Abstract

In this note we describe recent results on semiclassical random walk associated to a probability density which may also concentrate as the semiclassical parameter goes to zero. The main result gives a spectral asymptotics of the close to $1$ eigenvalues. This problem was studied in [1] and relies on a general factorization result for pseudo-differential operators. In this note we just sketch the proof of this second theorem. At the end of the note, using the factorization, we give a new proof of the spectral asymptotics based on some comparison argument.

Bibliographie

[1] J.-F. Bony, F. Hérau & L. Michel, “Tunnel effect for semiclassical random walks”, arXiv:1401.2935
[2] A. Bovier, V. Gayrard & M. Klein, “Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues”, J. Eur. Math. Soc. 7 (2005) no. 1, p. 69-99  MR 2120991 |  Zbl 1105.82025
[3] H. Cycon, R. Froese, W. Kirsch & B. Simon, Schrödinger operators with application to quantum mechanics and global geometry, Texts and Monographs in Physics, Springer-Verlag, 1987  MR 883643 |  Zbl 0619.47005
[4] M. Dimassi & J. Sjöstrand, Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series 268, Cambridge University Press, 1999  MR 1735654 |  Zbl 0926.35002
[5] B. Helffer, Semi-classical analysis for the Schrödinger operator and applications, Lecture Notes in Mathematics 1336, Springer-Verlag, 1988  MR 960278 |  Zbl 0647.35002
[6] B. Helffer, M. Klein & F. Nier, “Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach”, Mat. Contemp. 26 (2004), p. 41-85  MR 2111815 |  Zbl 1079.58025
[7] B. Helffer & F. Nier, Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians, Lecture Notes in Mathematics 1862, Springer-Verlag, 2005  MR 2130405 |  Zbl 1072.35006
[8] B. Helffer & J. Sjöstrand, “Puits multiples en mécanique semi-classique. IV. Étude du complexe de Witten”, Comm. Partial Differential Equations 10 (1985) no. 3, p. 245-340  MR 780068 |  Zbl 0597.35024
[9] F. Hérau, M. Hitrik & J. Sjöstrand, “Tunnel effect and symmetries for Kramers-Fokker-Planck type operators”, J. Inst. Math. Jussieu 10 (2011) no. 3, p. 567-634  MR 2806463 |  Zbl 1223.35246
[10] F. Hérau, M. Hitrik & J. Sjöstrand, “Supersymmetric structures for second order differential operators”, Algebra i Analiz 25 (2013) no. 2, p. 125-154  MR 3114853
[11] T. Lelièvre, M. Rousset & G. Stoltz, Free energy computations, Imperial College Press, 2010, A mathematical perspective  MR 2681239 |  Zbl 1227.82002
[12] A. Martinez, An introduction to semiclassical and microlocal analysis, Universitext, Springer-Verlag, 2002  MR 1872698 |  Zbl 0994.35003
[13] A. Martinez & M. Rouleux, “Effet tunnel entre puits dégénérés”, Comm. Partial Differential Equations 13 (1988) no. 9, p. 1157-1187  MR 946285 |  Zbl 0649.35073
[14] M. Reed & B. Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press, 1978  MR 493421 |  Zbl 0242.46001
[15] M. Zworski, Semiclassical analysis, Graduate Studies in Mathematics 138, American Mathematical Society, 2012  MR 2952218 |  Zbl 1252.58001
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