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

Harrell, Evans, II; Stubbe, Joachim

doi:10.1090/S0002-9947-2011-05252-9

Trace identities for commutators, with applications to the distribution of eigenvalues
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by Evans M. Harrell II and Joachim Stubbe

Trans. Amer. Math. Soc. 363 (2011), 6385-6405

DOI: https://doi.org/10.1090/S0002-9947-2011-05252-9

Published electronically: July 18, 2011

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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 $\|h\|_{\infty }^2 N^{2/d}$.

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