cnrs   Université de Toulouse
With cedram.org

Table of contents for this issue | Previous article | Next article
D. Le Peutrec
Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian
Annales de la faculté des sciences de Toulouse Sér. 6, 19 no. 3-4 (2010), p. 735-809, doi: 10.5802/afst.1265
Article PDF | Reviews MR 2790817 | Zbl 1213.58023

Résumé - Abstract

This article follows the previous works [HeKlNi, HeNi] by Helffer-Klein-Nier and Helffer-Nier about the metastability in reversible diffusion processes via a Witten complex approach. Again, exponentially small eigenvalues of some self-adjoint realization of $\Delta _{f,h}^{(0)}=-h^{2}\Delta +\left|\nabla f(x)\right|^{2}-h\Delta f(x)$ are considered as the small parameter $h>0$ tends to $0$. The function $f$ is assumed to be a Morse function on some bounded domain $\Omega $ with boundary $\partial \Omega $. Neumann type boundary conditions are considered. With these boundary conditions, some possible simplifications in the Dirichlet problem studied in [HeNi] are no more possible. A finer treatment of the three geometries involved in the boundary problem (boundary, metric, Morse function) is here carried out.

Bibliography

[Bis] Bismut (J.M.).— The Witten complex and the degenerate Morse inequalities. J. Differ. Geom. 23, p. 207-240 (1986).  MR 852155 |  Zbl 0608.58038
[BoEcGaKl] Bovier (A.), Eckhoff (M.), Gayrard (V.), and Klein (M.).— Metastability in reversible diffusion processes I: Sharp asymptotics for capacities and exit times. JEMS 6 (4), p. 399-424 (2004).  MR 2094397 |  Zbl 1076.82045
[BoGaKl] Bovier (A.), Gayrard (V.), and Klein (M.).— Metastability in reversible diffusion processes II: Precise asymptotics for small eigenvalues. JEMS 7 (1), p. 69-99 (2004).  MR 2120991 |  Zbl 1105.82025
[Bur] Burghelea (D.).— Lectures on Witten-Helffer-Sjöstrand theory. Gen. Math. 5, p. 85-99 (1997).  MR 1723597 |  Zbl 0936.58008
[ChLi] Chang (K.C.) and Liu (J.).— A cohomology complex for manifolds with boundary. Topological Methods in Non Linear Analysis, Vol. 5, p. 325-340 (1995).  MR 1374068 |  Zbl 0848.58001
[CyFrKiSi] Cycon (H.L), Froese (R.G), Kirsch (W.), and Simon (B.).— Schrödinger operators with application to quantum mechanics and global geometry. Text and Monographs in Physics, Springer Verlag, 2nd corrected printing (2008).  MR 883643 |  Zbl 0619.47005
[CoPaYc] Colin de Verdière (Y.), Pan (Y.), and Ycart (B.).— Singular limits of Schrödinger operators and Markov processes. J. Operator Theory 41, No. 1, p. 151-173 (1999).  MR 1675188 |  Zbl 0990.47013
[DiSj] Dimassi (M.) and Sjöstrand (J.).— Spectral Asymptotics in the semi-classical limit. London Mathematical Society, Lecture Note Series 268, Cambridge University Press (1999).  MR 1735654 |  Zbl 0926.35002
[Duf] Duff (G.F.D.).— Differential forms in manifolds with boundary. Ann. of Math. 56, p. 115-127 (1952).  MR 48136 |  Zbl 0049.18804
[DuSp] Duff (G.F.D.) and Spencer (D.C.).— Harmonic tensors on Riemannian manifolds with boundary. Ann. of Math. 56, p. 128-156 (1952).  MR 48137 |  Zbl 0049.18901
[FrWe] Freidlin (M.I.) and Wentzell (A.D.).— Random perturbations of dynamical systems. Transl. from the Russian by Joseph Szuecs. 2nd ed. Grundlehren der Mathematischen Wissenschaften, 260, New York (1998).  MR 1652127 |  Zbl 0522.60055
[GaHuLa] Gallot (S.), Hulin (D.), and Lafontaine (J.) Riemannian Geometry. Universitext, 2nd Edition, Springer Verlag (1993).  Zbl 0636.53001
[Gil] Gilkey (P.B.).— Invariance theory, the heat equation, and the Atiyah-Singer index theorem. Mathematics Lecture Series, 11, Publish or Perish, Wilmington (1984).  MR 783634 |  Zbl 0565.58035
[Gol] Goldberg (S.I.).— Curvature and Homology. Dover books in Mathematics, 3rd edition (1998).  MR 1635338 |  Zbl 0962.53001
[Gue] Guérini (P.) Prescription du spectre du Laplacien de Hodge-de Rham. Annales de l’ENS, Vol. 37 (2), p. 270-303 (2004). Numdam |  MR 2061782 |  Zbl 1068.58016
[Hel1] Helffer (B.).— Etude du Laplacien de Witten associé à une fonction de Morse dégénérée. Publications de l’université de Nantes, Séminaire EDP 1987-88.
[Hel2] Helffer (B.).— Introduction to the semi-classical Analysis for the Schrödinger operator and applications. Lecture Notes in Mathematics 1336, Springer Verlag (1988).  MR 960278 |  Zbl 0647.35002
[Hel3] Helffer (B.).— Semi-classical analysis, Witten Laplacians and statistical mechanics. World Scientific (2002).  Zbl 1046.82001
[Hen] Henniart (G.) .— Les inégalités de Morse (d’après E. Witten). Seminar Bourbaki, Vol. 1983/84, Astérisque No. 121-122, p. 43-61 (1985). Numdam |  MR 768953 |  Zbl 0565.58033
[HeKlNi] Helffer (B.), Klein (M.), and Nier (F.).— Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach. Matematica Contemporanea, 26, p. 41-85 (2004).  MR 2111815 |  Zbl 1079.58025
[HeNi] Helffer (B.) and Nier (F.).— Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary. Mémoire 105, Société Mathématique de France (2006).  MR 2270650 |  Zbl 1108.58018
[HeSj1] Helffer (B.) and Sjöstrand (J.).— Multiple wells in the semi-classical limit I. Comm. Partial Differential Equations 9 (4), p. 337-408 (1984).  MR 740094 |  Zbl 0546.35053
[HeSj2] Helffer (B.) and Sjöstrand (J.).— Puits multiples en limite semi-classique II -Interaction moléculaire-Symétries-Perturbations. Ann. Inst. H. Poincaré Phys. Théor. 42 (2), p. 127-212 (1985). Numdam |  MR 798695 |  Zbl 0595.35031
[HeSj4] Helffer (B.) and Sjöstrand (J.).— Puits multiples en limite semi-classique IV -Etude du complexe de Witten -. Comm. Partial Differential Equations 10 (3), p. 245-340 (1985).  MR 780068 |  Zbl 0597.35024
[HeSj5] Helffer (B.) and Sjöstrand (J.).— Puits multiples en limite semi-classique V - Etude des minipuits-. Current topics in partial differential equations, p. 133-186, Kinokuniya, Tokyo (1986).  MR 1112146 |  Zbl 0628.35024
[HoKuSt] Holley (R.), Kusuoka (S.), and Stroock (D.).— Asymptotics of the spectral gap with applications to the theory of simulated annealing. J. Funct. Anal. 83 (2), p. 333-347 (1989).  MR 995752 |  Zbl 0706.58075
[Kol] Kolokoltsov (V.N.).— Semi-classical analysis for diffusions and stochastic processes. Lecture Notes in Mathematics 1724, Springer Verlag (2000).  MR 1755149 |  Zbl 0951.60001
[KoMa] Kolokoltsov (V.N.), and Makarov (K.).— Asymptotic spectral analysis of a small diffusion operator and the life times of the corresponding diffusion process. Russian J. Math. Phys. 4 (3), p. 341-360 (1996).  MR 1443178 |  Zbl 0912.58042
[KoPrSh] Koldan (N.), Prokhorenkov (I.), and Shubin (M.).— Semiclassical Asymptotics on Manifolds with Boundary. Preprint (2008). http://arxiv.org/abs/0803.2502v1  MR 1500151
[Lau] Laudenbach (F.).— Topologie différentielle. Cours de Majeure de l’Ecole Polytechnique (1993).
[Lep1] Le Peutrec (D.).— Small singular values of an extracted matrix of a Witten complex. Cubo, A Mathematical Journal, Vol. 11 (4), p. 49-57 (2009).  MR 2571794 |  Zbl 1181.81050
[Lep2] Le Peutrec (D.).— Local WKB construction for Witten Laplacians on manifolds with boundary. Analysis & PDE, Vol. 3, No. 3, p. 227-260 (2010).  MR 2672794
[Mic] Miclo (L.).— Comportement de spectres d’opérateurs à basse température. Bull. Sci. Math. 119, p. 529-533 (1995).  MR 1364276 |  Zbl 0840.60057
[Mil1] Milnor (J.W.).— Morse Theory. Princeton University press (1963).  MR 163331 |  Zbl 0108.10401
[Mil2] Milnor (J.W.).— Lectures on the $h$-cobordism Theorem. Princeton University press (1965).  MR 190942 |  Zbl 0161.20302
[Nie] Nier (F.).— Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach. Journées “Equations aux Dérivées Partielles”, Exp No VIII, Ecole Polytechnique (2004). Cedram |  MR 2135363 |  Zbl 1067.35057
[Per] Persson (A.).— Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator. Math. Scandinavica 8, p. 143-153 (1960).  MR 133586 |  Zbl 0145.14901
[Sch] Schwarz (G.).— Hodge decomposition. A method for Solving Boundary Value Problems. Lecture Notes in Mathematics 1607, Springer Verlag (1995).  MR 1367287 |  Zbl 0828.58002
[Sima] Simader (C.G.).— Essential self-adjointness of Schrödinger operators bounded from below. Math. Z. 159, p. 47-50 (1978).  MR 470456 |  Zbl 0409.35026
[Sim] Simon (B.).— Semi-classical analysis of low lying eigenvalues, I. Nondegenerate minima: Asymptotic expansions. Ann. Inst. H. Poincaré, Phys. Théor. 38, p. 296-307 (1983). Numdam |  MR 708966 |  Zbl 0526.35027
[Wit] Witten (E.).— Supersymmetry and Morse inequalities. J. Diff. Geom. 17, p. 661-692 (1982).  MR 683171 |  Zbl 0499.53056
[Zha] Zhang (W.).— Lectures on Chern-Weil theory and Witten deformations. Nankai Tracts in Mathematics, Vol. 4, World Scientific (2002).  MR 1864735 |  Zbl 0993.58014
Search for an article
Search within the site
top