Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian

D. Le Peutrec

doi:10.5802/afst.1265

D. Le Peutrec ¹

¹ Laboratoire de Mathématiques, UMR-CNRS 8628, Université Paris-Sud 11, Bâtiment 425, 91405 Orsay, France

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. 3-4, pp. 735-809.

Abstract
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 $Δ_{f, h}^{(0)} = - h^{2} Δ + {|\nabla f (x)|}^{2} - h Δ f (x)$ 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 $Ω$ with boundary $\partial Ω$ . 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.

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 $Δ_{f, h}^{(0)} = - h^{2} Δ + {|\nabla f (x)|}^{2} - h Δ f (x)$ lorsque le paramètre $h > 0$ tend vers $0$ . La fonction $f$ est une fonction de Morse sur un domaine borné $Ω$ de bord $\partial Ω$ . 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.

EuDML MR Zbl

DOI: 10.5802/afst.1265

Author's affiliations:

D. Le Peutrec ¹

¹ Laboratoire de Mathématiques, UMR-CNRS 8628, Université Paris-Sud 11, Bâtiment 425, 91405 Orsay, France

@article{AFST_2010_6_19_3-4_735_0,
     author = {D. Le Peutrec},
     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 = {https://afst.centre-mersenne.org/articles/10.5802/afst.1265/}
}

TY  - JOUR
AU  - D. Le Peutrec
TI  - Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2010
SP  - 735
EP  - 809
VL  - 19
IS  - 3-4
PB  - Université Paul Sabatier, Institut de Mathématiques
PP  - Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1265/
DO  - 10.5802/afst.1265
LA  - en
ID  - AFST_2010_6_19_3-4_735_0
ER  -

%0 Journal Article
%A D. Le Peutrec
%T Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2010
%P 735-809
%V 19
%N 3-4
%I Université Paul Sabatier, Institut de Mathématiques
%C Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1265/
%R 10.5802/afst.1265
%G en
%F AFST_2010_6_19_3-4_735_0

D. Le Peutrec. Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. 3-4, pp. 735-809. doi : 10.5802/afst.1265. https://afst.centre-mersenne.org/articles/10.5802/afst.1265/

References
Cited by

[Bis] Bismut (J.M.).— The Witten complex and the degenerate Morse inequalities. J. Differ. Geom. 23, p. 207-240 (1986). | MR | Zbl

[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 | Zbl

[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 | Zbl

[Bur] Burghelea (D.).— Lectures on Witten-Helffer-Sjöstrand theory. Gen. Math. 5, p. 85-99 (1997). | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[Duf] Duff (G.F.D.).— Differential forms in manifolds with boundary. Ann. of Math. 56, p. 115-127 (1952). | MR | Zbl

[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 | Zbl

[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 | Zbl

[GaHuLa] Gallot (S.), Hulin (D.), and Lafontaine (J.) Riemannian Geometry. Universitext, 2nd Edition, Springer Verlag (1993). | Zbl

[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 | Zbl

[Gol] Goldberg (S.I.).— Curvature and Homology. Dover books in Mathematics, 3rd edition (1998). | MR | Zbl

[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 | Zbl

[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 | Zbl

[Hel3] Helffer (B.).— Semi-classical analysis, Witten Laplacians and statistical mechanics. World Scientific (2002). | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[Kol] Kolokoltsov (V.N.).— Semi-classical analysis for diffusions and stochastic processes. Lecture Notes in Mathematics 1724, Springer Verlag (2000). | MR | Zbl

[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 | Zbl

[KoPrSh] Koldan (N.), Prokhorenkov (I.), and Shubin (M.).— Semiclassical Asymptotics on Manifolds with Boundary. Preprint (2008). http://arxiv.org/abs/0803.2502v1 | MR

[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 | Zbl

[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

[Mic] Miclo (L.).— Comportement de spectres d’opérateurs à basse température. Bull. Sci. Math. 119, p. 529-533 (1995). | MR | Zbl

[Mil1] Milnor (J.W.).— Morse Theory. Princeton University press (1963). | MR | Zbl

[Mil2] Milnor (J.W.).— Lectures on the $h$ -cobordism Theorem. Princeton University press (1965). | MR | Zbl

[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 | Zbl

[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 | Zbl

[Sch] Schwarz (G.).— Hodge decomposition. A method for Solving Boundary Value Problems. Lecture Notes in Mathematics 1607, Springer Verlag (1995). | MR | Zbl

[Sima] Simader (C.G.).— Essential self-adjointness of Schrödinger operators bounded from below. Math. Z. 159, p. 47-50 (1978). | MR | Zbl

[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 | Zbl

[Wit] Witten (E.).— Supersymmetry and Morse inequalities. J. Diff. Geom. 17, p. 661-692 (1982). | MR | Zbl

[Zha] Zhang (W.).— Lectures on Chern-Weil theory and Witten deformations. Nankai Tracts in Mathematics, Vol. 4, World Scientific (2002). | MR | Zbl

Cited by Sources: