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Trace identities for commutators, with applications to the distribution of eigenvalues

Authors: Evans M. Harrell II and Joachim Stubbe
Journal: Trans. Amer. Math. Soc. 363 (2011), 6385-6405
MSC (2010): Primary 81Q10, 35J25, 35P15, 35P20, 58C40
Published electronically: July 18, 2011
MathSciNet review: 2833559
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Abstract: We prove trace identities for commutators of operators, which are used to derive sum rules and sharp universal bounds for the eigenvalues of periodic Schrödinger operators and Schrödinger operators on immersed manifolds. In particular, we prove bounds on the eigenvalue $ \lambda_{N+1}$ in terms of the lower spectrum, bounds on ratios of means of eigenvalues, and universal monotonicity properties of eigenvalue moments, which imply sharp versions of Lieb-Thirring inequalities. In the case of a Schrödinger operator on an immersed manifold of dimension $ d$, we derive a version of Reilly's inequality bounding the eigenvalue $ \lambda_{N+1}$ of the Laplace-Beltrami operator by a universal constant times $ \Vert h\Vert _{\infty}^2 N^{2/d}$.

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Additional Information

Evans M. Harrell II
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0610

Joachim Stubbe
Affiliation: Department of Mathematics, Ecole Polytechnique Federale de Lausanne, IMB-FSB, Station 8, CH-1015 Lausanne, Switzerland

Received by editor(s): November 11, 2009
Published electronically: July 18, 2011
Article copyright: © Copyright 2011 by the authors

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