On riesz means of eigenvalues

Evans M. Harrell, Lotfi Hermi

Research output: Contribution to journalArticle

8 Scopus citations

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 monoton ic 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.

Original languageEnglish (US)
Pages (from-to)1521-1543
Number of pages23
JournalCommunications in Partial Differential Equations
Volume36
Issue number9
DOIs
StatePublished - Sep 1 2011

Keywords

  • Berezin-li-yau inequalities
  • Eigenvalues
  • Laplacian
  • Universal bounds

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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