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Mathematics > Spectral Theory

arXiv:0712.4088 (math)
[Submitted on 25 Dec 2007]

Title:On Riesz Means of Eigenvalues

Authors:Evans M. Harrell, Lotfi Hermi
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Abstract: In this article we prove the equivalence of certain inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian with a classical inequality of Kac. Connections are made via integral transforms including those of Laplace, Legendre, Weyl, and Mellin, and the Riemann-Liouville fractional transform. We also prove new universal eigenvalue inequalities and monotonicity principles for Dirichlet Laplacians as well as certain Schrödinger operators. At the heart of these inequalities are calculations of commutators of operators, sum rules, and monotonic properties of Riesz means. In the course of developing these inequalities we prove new bounds for the partition function and the spectral zeta function (cf. Corollaries 3.5-3.7) and conjecture about additional bounds.
Comments: 26 pages; 3 figures; submitted
Subjects: Spectral Theory (math.SP); Mathematical Physics (math-ph)
MSC classes: 35P15; 47A75; 49R50; 58J50
Cite as: arXiv:0712.4088 [math.SP]
  (or arXiv:0712.4088v1 [math.SP] for this version)
  https://doi.org/10.48550/arXiv.0712.4088

Submission history

From: Lotfi Hermi [view email]
[v1] Tue, 25 Dec 2007 23:59:47 UTC (159 KB)
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