Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 19 (2010) no. 3-4, pp. 735-809.

Cet article est dans la continuation des travaux [HeKlNi, HeNi] de Helffer-Klein-Nier et Helffer-Nier sur l’étude de la métastabilité dans des processus de diffusions réversibles via une approche de Witten. Nous considérons encore ici les valeurs propres exponentionnellement petites d’une réalisation auto-adjointe de Δ f,h (0) =-h 2 Δ+f(x) 2 -hΔf(x) lorsque le paramètre h>0 tend vers 0. La fonction f est une fonction de Morse sur un domaine borné Ω de bord Ω. Des conditions au bord de type Neumann sont considérées ici. Avec ces conditions, certaines simplifications utilisées pour l’étude du problème de Dirichlet dans [HeNi] ne sont plus possibles. Un traitement plus fin des trois géométries intervenant dans le problème à bord (bord, métrique, fonction de Morse) est donc nécessaire.

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 Δ f,h (0) =-h 2 Δ+f(x) 2 -hΔf(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 Ω with boundary Ω. 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.

@article{AFST_2010_6_19_3-4_735_0,
     author = {Le Peutrec, D.},
     title = {Small eigenvalues of the {Neumann} realization of the semiclassical {Witten} {Laplacian}},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {735--809},
     publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 19},
     number = {3-4},
     year = {2010},
     doi = {10.5802/afst.1265},
     mrnumber = {2790817},
     zbl = {1213.58023},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/afst.1265/}
}
TY  - JOUR
AU  - Le Peutrec, D.
TI  - Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2010
DA  - 2010///
SP  - 735
EP  - 809
VL  - Ser. 6, 19
IS  - 3-4
PB  - Université Paul Sabatier, Institut de mathématiques
PP  - Toulouse
UR  - http://www.numdam.org/articles/10.5802/afst.1265/
UR  - https://www.ams.org/mathscinet-getitem?mr=2790817
UR  - https://zbmath.org/?q=an%3A1213.58023
UR  - https://doi.org/10.5802/afst.1265
DO  - 10.5802/afst.1265
LA  - en
ID  - AFST_2010_6_19_3-4_735_0
ER  - 
Le Peutrec, D. Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 19 (2010) no. 3-4, pp. 735-809. doi : 10.5802/afst.1265. http://www.numdam.org/articles/10.5802/afst.1265/

[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). | Numdam | 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). | Numdam | 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

Cité par Sources :