A unified approach to universal inequalities for eigenvalues of elliptic operators

Mark S. Ashbaugh, Lotfi Hermi

Research output: Contribution to journalArticle

32 Citations (Scopus)

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
StatePublished - Dec 2004

Fingerprint

Elliptic Operator
Eigenvalue
Operator
Discrete Spectrum
Strengthening
Self-adjoint Operator
Commutator
Rayleigh
Dirichlet Boundary Conditions
Algebra
Alternatives

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

A unified approach to universal inequalities for eigenvalues of elliptic operators. / Ashbaugh, Mark S.; Hermi, Lotfi.

In: Pacific Journal of Mathematics, Vol. 217, No. 2, 12.2004, p. 201-219.

Research output: Contribution to journalArticle

@article{13ba1a11bcfc473e897ba647a9c71354,
title = "A unified approach to universal inequalities for eigenvalues of elliptic operators",
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-P{\'o}lya-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.",
author = "Ashbaugh, {Mark S.} and Lotfi Hermi",
year = "2004",
month = "12",
language = "English (US)",
volume = "217",
pages = "201--219",
journal = "Pacific Journal of Mathematics",
issn = "0030-8730",
publisher = "University of California, Berkeley",
number = "2",

}

TY - JOUR

T1 - A unified approach to universal inequalities for eigenvalues of elliptic operators

AU - Ashbaugh, Mark S.

AU - Hermi, Lotfi

PY - 2004/12

Y1 - 2004/12

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=12744260598&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=12744260598&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:12744260598

VL - 217

SP - 201

EP - 219

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 2

ER -