Pollicott-Ruelle spectrum and Witten Laplacians
Document type :
Compte-rendu et recension critique d'ouvrage
DOI :
Title :
Pollicott-Ruelle spectrum and Witten Laplacians
Author(s) :
Dang, Nguyen Viet [Auteur]
Probabilités, statistique, physique mathématique [PSPM]
Riviere, Gabriel [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Probabilités, statistique, physique mathématique [PSPM]
Riviere, Gabriel [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Journal title :
Journal of the European Mathematical Society
Publisher :
European Mathematical Society
Publication date :
2019
ISSN :
1435-9855
English keyword(s) :
microlocal analysis
resonances
Morse-Smale gradient flows
hyperbolic dynamical systems
differential topology
resonances
Morse-Smale gradient flows
hyperbolic dynamical systems
differential topology
HAL domain(s) :
Mathematics [math]/Dynamical Systems [math.DS]
Mathématiques [math]/Théorie spectrale [math.SP]
Mathématiques [math]/Physique mathématique [math-ph]
Mathématiques [math]/Equations aux dérivées partielles [math.AP]
Mathématiques [math]/Théorie spectrale [math.SP]
Mathématiques [math]/Physique mathématique [math-ph]
Mathématiques [math]/Equations aux dérivées partielles [math.AP]
English abstract : [en]
We study the asymptotic behaviour of eigenvalues and eigenmodes of the Witten Laplacian on a smooth compact Riemannian manifold without boundary. We show that they converge to the Pollicott-Ruelle spectrum of the corresponding ...
Show more >We study the asymptotic behaviour of eigenvalues and eigenmodes of the Witten Laplacian on a smooth compact Riemannian manifold without boundary. We show that they converge to the Pollicott-Ruelle spectrum of the corresponding gradient flow acting on appropriate anisotropic Sobolev spaces. In particular, our results relate the approach of Laudenbach and Harvey–Lawson to Morse theory using currents, which was discussed in previous work of the authors, and Witten's point of view based on semiclassical analysis and tunneling. As an application of our methods, we also construct a natural family of quasimodes satisfying the Witten-Helffer-Sjöstrand tunneling formulas and the Fukaya conjecture on Witten deformation of the wedge product.Show less >
Show more >We study the asymptotic behaviour of eigenvalues and eigenmodes of the Witten Laplacian on a smooth compact Riemannian manifold without boundary. We show that they converge to the Pollicott-Ruelle spectrum of the corresponding gradient flow acting on appropriate anisotropic Sobolev spaces. In particular, our results relate the approach of Laudenbach and Harvey–Lawson to Morse theory using currents, which was discussed in previous work of the authors, and Witten's point of view based on semiclassical analysis and tunneling. As an application of our methods, we also construct a natural family of quasimodes satisfying the Witten-Helffer-Sjöstrand tunneling formulas and the Fukaya conjecture on Witten deformation of the wedge product.Show less >
Language :
Anglais
Popular science :
Non
Comment :
Revised version following referees suggestions. To appear J. Eur. Math. Soc.
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