A unified approach to universal inequalities for eigenvalues of elliptic operators

Mark S. Ashbaugh, Lotfi Hermi

Research output: Contribution to journalArticle

32 Scopus citations

Abstract

We present an abstract approach to universal inequalities for the discrete spectrum of a self-adjoint operator, based on commutator algebra, the Rayleigh-Ritz principle, and one set of "auxiliary" operators. The new proof unifies classical inequalities of Payne-Pólya-Weinberger, Hile-Protter, and H.C. Yang and provides a Yang type strengthening of Hook's bounds for various elliptic operators with Dirichlet boundary conditions. The proof avoids the introduction of the "free parameters" of many previous authors and relies on earlier works of Ashbaugh and Benguria, and, especially, Harrell (alone and with Michel), in addition to those of the other authors listed above. The Yang type inequality is proved to be stronger under general conditions on the operator and the auxiliary operators. This approach provides an alternative route to recent results obtained by Harrell and Stubbe.

Original languageEnglish (US)
Pages (from-to)201-219
Number of pages19
JournalPacific Journal of Mathematics
Volume217
Issue number2
DOIs
StatePublished - Dec 2004

ASJC Scopus subject areas

  • Mathematics(all)

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