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
U2 - 10.2140/pjm.2004.217.201
DO - 10.2140/pjm.2004.217.201
M3 - Article
AN - SCOPUS:12744260598
SN - 0030-8730
VL - 217
SP - 201
EP - 219
JO - Pacific Journal of Mathematics
JF - Pacific Journal of Mathematics
IS - 2
ER -