Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds
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- by Ahmad El Soufi, Evans M. Harrell II and Saïd Ilias PDF
- Trans. Amer. Math. Soc. 361 (2009), 2337-2350
Abstract:
We establish inequalities for the eigenvalues of Schrödinger operators on compact submanifolds (possibly with nonempty boundary) of Euclidean spaces, of spheres, and of real, complex and quaternionic projective spaces, which are related to inequalities for the Laplacian on Euclidean domains due to Payne, Pólya, and Weinberger and to Yang, but which depend in an explicit way on the mean curvature. In later sections, we prove similar results for Schrödinger operators on homogeneous Riemannian spaces and, more generally, on any Riemannian manifold that admits an eigenmap into a sphere, as well as for the Kohn Laplacian on subdomains of the Heisenberg group.
Among the consequences of this analysis are an extension of Reilly’s inequality, bounding any eigenvalue of the Laplacian in terms of the mean curvature, and spectral criteria for the immersibility of manifolds in homogeneous spaces.
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Additional Information
- Ahmad El Soufi
- Affiliation: Laboratoire de Mathématiques et Physique Théorique, Université François Rabelais de Tours, UMR-CNRS 6083, Parc de Grandmont, 37200 Tours, France
- Email: elsoufi@univ-tours.fr
- Evans M. Harrell II
- Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
- MR Author ID: 81525
- Email: harrell@math.gatech.edu
- Saïd Ilias
- Affiliation: Laboratoire de Mathématiques et Physique Théorique, Université François Rabelais de Tours, UMR-CNRS 6083, Parc de Grandmont, 37200 Tours, France
- Email: ilias@univ-tours.fr
- Received by editor(s): January 16, 2007
- Published electronically: December 16, 2008
- © Copyright 2008 by the authors
- Journal: Trans. Amer. Math. Soc. 361 (2009), 2337-2350
- MSC (2000): Primary 58J50, 58E11, 35P15
- DOI: https://doi.org/10.1090/S0002-9947-08-04780-6
- MathSciNet review: 2471921