Spectral approximation by \(L_ 2\) discretization of the Laplace operator on manifolds of bounded geometry
\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 \(\sigma_ e(\Delta)\).
scientific article; zbMATH DE number 4102155
scientific article