Spectral approximation by \(L_ 2\) discretization of the Laplace operator on manifolds of bounded geometry
type: Item
instance of: scholarly article
title: Spectral approximation by \(L_ 2\) discretization of the Laplace operator on manifolds of bounded geometry
zbMATH Open document ID: 0673.58049
DOI: 10.1007/BF00054610
author: Ingolf Buttig
published in: Annals of Global Analysis and Geometry
publication date: 1988-01-01T00:00:00Z
review text: \textit{J. Dodziuk} and \textit{V. K. Patodi} [J. Indian Math. Soc., New Ser. 40, 1-52 (1976; Zbl 0435.58004)] proved that in a closed oriented Riemannian manifold M, all the eigenvalues and eigenforms of the Friedrich's extension \(\Delta_ F\) of the analytic Laplacian of M can be approximated by eigenvalues and eigencochains of the semicombinatorial Laplacian. This operator is obtained by the Whitney map, whose left inverse is the de Rham map associated to a smooth triangulation of M. In the case of an open complete Riemannian manifold of bounded geometry and adapting Dodziuk-Patodi's proof, the author approximates the eigenvalues of the Laplacian \(\Delta\) below the essential spectrum \(\sigma_ e(\Delta)\) together with their eigenforms. In one of the two Appendices of the paper the author presents an example of a manifold of bounded geometry which admits discrete eigenvalues below \(\sigma_ e(\Delta).\) On the other hand, \textit{J. Eichhorn} [Spektrum, Geometrie und Topologie offener Riemannscher Mannigfaltigkeiten (to appear)] established that in a manifold of bounded geometry there exists a ray contained in \(\sigma_ e(\Delta)\). The author also derives a similar result about approximation concerning the ray contained in \(\sigma_ e(\Delta)\).
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 4102155
zbMATH Keywords: semicombinatorical Laplacian
zbMATH Keywords: de Rham map
zbMATH Keywords: Hodge decomposition of
zbMATH Keywords: Whitney forms
zbMATH Keywords: the spectrum of the Laplacian
zbMATH Keywords: Whitney map
zbMATH Keywords: Laplacian
reviewed by: Enrique Outerelo
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00054610
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description: scientific article; zbMATH DE number 4102155