This article is a continuation of previous works by Bovier-Eckhoff-Gayrard-Klein, Bovier-Gayrard-Klein and Helffer-Klein-Nier. The main object is the analysis of the small eigenvalues (as ) of the Laplacian attached to the quadratic form , where is a bounded connected open set with -boundary and is a Morse function on . The previous works were devoted to the case of a manifold which is compact but without boundary or . Our aim is here to analyze the case with boundary. After the introduction of a Witten cohomology complex adapted to the case with boundary, we give a very accurate asymptotics for the exponentially small eigenvalues. In particular, we analyze the effect of the boundary in the asymptotics.
Cet article prolonge des travaux antérieurs de Bovier-Eckhoff-Gayrard-Klein, Bovier-Gayrard-Klein et Helffer-Klein-Nier. L’objet principal en est l’analyse des petites valeurs propres du Laplacien associé à la forme quadratique , où est un domaine borné régulier et est une fonction de Morse sur . Les travaux précédents traitaient le cas d’une variété compacte sans bord ou le cas . Ici nous analysons le cas d’une variété compacte à bord. Après l’introduction d’un complexe de cohomologie de Witten adapté au cas à bord, nous donnons une description très précise des valeurs propres exponentiellement petites. En particulier, nous traitons l’effet du bord sur les développements asymptotiques.
Keywords: Witten complex, Semiclassical expansion, exponentially small quantities, manifolds with boundary
Mot clés : Complexe de Witten, Développements semiclassiques, valeurs propres exponentiellement petites, variétés à bord
@book{MSMF_2006_2_105__1_0, author = {Helffer, Bernard and Nier, Francis}, title = {Quantitative analysis of metastability in reversible diffusion processes via a {Witten} complex approach: the case with boundary}, series = {M\'emoires de la Soci\'et\'e Math\'ematique de France}, publisher = {Soci\'et\'e math\'ematique de France}, number = {105}, year = {2006}, doi = {10.24033/msmf.417}, mrnumber = {2270650}, zbl = {1108.58018}, language = {en}, url = {http://www.numdam.org/item/MSMF_2006_2_105__1_0/} }
TY - BOOK AU - Helffer, Bernard AU - Nier, Francis TI - Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary T3 - Mémoires de la Société Mathématique de France PY - 2006 IS - 105 PB - Société mathématique de France UR - http://www.numdam.org/item/MSMF_2006_2_105__1_0/ DO - 10.24033/msmf.417 LA - en ID - MSMF_2006_2_105__1_0 ER -
%0 Book %A Helffer, Bernard %A Nier, Francis %T Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary %S Mémoires de la Société Mathématique de France %D 2006 %N 105 %I Société mathématique de France %U http://www.numdam.org/item/MSMF_2006_2_105__1_0/ %R 10.24033/msmf.417 %G en %F MSMF_2006_2_105__1_0
Helffer, Bernard; Nier, Francis. Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary. Mémoires de la Société Mathématique de France, Serie 2, no. 105 (2006), 95 p. doi : 10.24033/msmf.417. http://numdam.org/item/MSMF_2006_2_105__1_0/
[1] « The Witten complex and the degenerate Morse inequalities », J. Differ. Geom. 23 (1986), p. 207–240. | MR | Zbl
–[2] « Application of semi-classical analysis to the asymptotic study of the supercooling field of a superconducting material », Ann. Inst. H. Poincaré Phys. Théor. 58 (1991), no. 2, p. 189–233. | MR | EuDML | Zbl | Numdam
& –[3] « Metastability in reversible diffusion processes I: Sharp asymptotics for capacities and exit times », JEMS 6 (2004), no. 4, p. 399–424. | MR | EuDML | Zbl
, , & –[4] « Metastability in reversible diffusion processes II: Precise asymptotics for small eigenvalues », JEMS 7 (2004), no. 1, p. 69–99. | MR | EuDML | Zbl
, & –[5] « Lectures on Witten-Helffer-Sjöstrand theory », Gen. Math. 5 (1997), p. 85–99. | MR | EuDML | Zbl
–[6] « A cohomology complex for manifolds with boundary », Topol. Methods in Nonlinear Anal. 5 (1995), no. 2, p. 325–340. | MR | Zbl
& –[7] Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, Text and Monographs in Physics, Springer-Verlag, 1987. | MR | Zbl
, , & –[8] Elliptic boundary value problems on corner domains. Smoothness and asymptotic solutions, Lect. Notes in Math., vol. 1341, Springer Verlag, 1988. | MR | Zbl
–[9] Spectral asymptotics in the semiclassical limit, London Mathematical Society Lecture Note Series, vol. 268, Cambridge University Press, 1999. | MR
& –[10] Random perturbations of dynamical systems. 2nd edition, 2nd éd., Grundlehren der Mathematischen Wissenschaften, vol. 260, Springer Verlag, 1998. | MR
& –[11] Riemannian Geometry, 2nd éd., Universitext, Springer Verlag, 1993. | MR | Zbl
, & –[12] Introduction à la théorie des opérateurs linéaires non auto-adjoints dans un espace hilbertien, Monographies Universitaires de Mathématiques, vol. 39, Dunod, 1971. | MR
& –[13] Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman, 1985. | MR | Zbl
–[14] « Prescription du spectre du Laplacien de Hodge-de Rham », Ann. ENS 37 (2004), no. 2, p. 270–303. | MR | EuDML | Zbl | Numdam
–[15] « Etude du laplacien de Witten associé à une fonction de Morse dégénérée », Publications de l’université de Nantes, Séminaire EDP 1987-88.
–[16] —, Semi-classical analysis for the Schrödinger operator and applications, Lect. Notes in Mathematics, vol. 1336, Springer Verlag, Berlin, 1988.
[17] —, Semi-classical analysis, Witten Laplacians and statistical mechanics, World Scientific, 2002.
[18] « Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach. », Matematica Contemporanea 26 (2004), p. 41–85. | MR | Zbl
, & –[19] « Multiple wells in the semi-classical limit I », Comm. Partial Differential Equations. 9 (1984), no. 4, p. 337–408. | MR | Zbl
& –[20] —, « Puits multiples en limite semi-classique II-interaction moléculaire-symétries-perturbations », Ann. Inst. H. Poincaré Phys. Théor. 42 (1985), no. 2, p. 127–212. | MR | EuDML | Zbl | Numdam
[21] —, « Puits multiples en limite semi-classique IV-étude du complexe de Witten - », Comm. Partial Differential Equations 10 (1985), no. 3, p. 245–340. | MR | Zbl
[22] —, « Puits multiples en limite semi-classique V - Etude des minipuits - », in Current topics in partial differential equations, 1986. | Zbl
[23] « Asymptotics of the spectral gap with applications to the theory of simulated annealing », J. Funct. Anal. 83 (1989), no. 2, p. 333–347. | MR | Zbl
, & –[24] « On the spectral properties of Witten Laplacians, their range projections and Brascamp-Lieb’s inequality », Integral Equations and Operator Theory 36 (2000), no. 2, p. 288–324. | MR | Zbl
–[25] Semiclassical analysis for diffusions and stochastic processes, Lect. Notes in Mathematics, vol. 1724, Springer Verlag, 2000. | MR | Zbl
–[26] « Asymptotic spectral analysis of a small diffusion operator and the life times of the corresponding diffusion process », Russian J. Math. Phys. 4 (1996), no. 3, p. 341–360. | MR | Zbl
& –[27] « Boundary value problems for elliptic equations in domains with conical or angular points », Trudy Moskov. Mat. Obshch. 16 (1967), p. 209–292. | MR
–[28] « Comportement de spectres d’opérateurs à basse température », Bull. Sci. Math. 119 (1995), p. 529–533. | MR
–[29] Hodge decomposition. A method for Solving Boundary Value Problems, Lect. Notes Series, vol. 1607, Springer Verlag, 1995. | MR
–[30] « Essential self-adjointness of schrödinger operators bounded from below », Math. Z. 159 (1978), p. 47–50. | MR | EuDML | Zbl
–[31] Trace ideals and their applications., Lecture Note Series, vol. 35, London Mathematical Society, Cambridge University Press. IX, 1979. | MR | Zbl
–[32] —, « Semi-classical analysis of low lying eigenvalues, I. Nondegenerate minima: Asymptotic expansions », Ann. Inst. H. Poincaré Phys. Théor. 38 (1983), p. 296–307. | EuDML
[33] « Supersymmetry and Morse inequalities », J. Differ. Geom. 17 (1982), p. 661–692. | MR | Zbl
–[34] Lectures on Chern-Weil theory and Witten deformations, Nankai Tracts in Mathematics, vol. 4, World Scientific, 2002. | MR
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