On Harrell-Stubbe Type Inequalities for the Discrete Spectrum of a Self-Adjoint Operator

Details On Harrell-Stubbe Type Inequalities for the Discrete Spectrum of a Self-Adjoint Operator On Harrell-Stubbe Type Inequalities for the Discrete Spectrum of a Self-Adjoint Operator

2007-12-28

arxiv.org/abs/0712.4396v1

Mathematics

Spectral Theory

42 pages

Scientific paper

We produce a new proof and extend results by Harrell and Stubbe for the discrete spectrum of a self-adjoint operator. An abstract approach--based on commutator algebra, the Rayleigh-Ritz principle, and an ``optimal'' usage of the Cauchy-Schwarz inequality--is used to produce ``parameter-free'', ``projection-free'' versions of their theorems. We also analyze the strength of the various inequalities that ensue. The results contain classical bounds for the eigenvalues. Extensions of a variety of inequalities \`a la Harrell-Stubbe are illustrated for both geometric and physical problems.

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