Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian

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 (2010)

  • Volume: 19, Issue: 3-4, page 735-809
  • ISSN: 0240-2963

Abstract

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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 - 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 ∂ 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.

How to cite

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Le Peutrec, D.. "Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian." Annales de la faculté des sciences de Toulouse Mathématiques 19.3-4 (2010): 735-809. <http://eudml.org/doc/115880>.

@article{LePeutrec2010,
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 $h&gt;0$ 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.},
affiliation = {Laboratoire de Mathématiques, UMR-CNRS 8628, Université Paris-Sud 11, Bâtiment 425, 91405 Orsay, France},
author = {Le Peutrec, D.},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {Friedrich extension; quasi-mode; WKB construction; boundary complex; localization of the spectrum; Morse function},
language = {eng},
number = {3-4},
pages = {735-809},
publisher = {Université Paul Sabatier, Toulouse},
title = {Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian},
url = {http://eudml.org/doc/115880},
volume = {19},
year = {2010},
}

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
PB - Université Paul Sabatier, Toulouse
VL - 19
IS - 3-4
SP - 735
EP - 809
AB - 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 $h&gt;0$ 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.
LA - eng
KW - Friedrich extension; quasi-mode; WKB construction; boundary complex; localization of the spectrum; Morse function
UR - http://eudml.org/doc/115880
ER -

References

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