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Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds
Author(s):
Ahmad
El Soufi;
Evans
M.
Harrell II;
Saïd
Ilias
Journal:
Trans. Amer. Math. Soc.
361
(2009),
2337-2350.
MSC (2000):
Primary 58J50, 58E11, 35P15
Posted:
December 16, 2008
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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
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
DOI:
10.1090/S0002-9947-08-04780-6
PII:
S 0002-9947(08)04780-6
Keywords:
Spectrum,
eigenvalue,
Laplacian,
Schr\"{o}dinger operator,
Reilly inequality,
Kohn Laplacian.
Received by editor(s):
January 16, 2007
Posted:
December 16, 2008
Copyright of article:
Copyright
2008,
by the authors
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