For the eigenvalues of a Schroedinger operator with a nonnegative potential, Dolbeault, Felmer, Loss, and Paturel proved (JFA, 2006) a class of inequalities of Lieb-Thirring type relating functions of the spectrum to (appropriate) integrals which depend on the potential. These inequalities include in particular the classical Golden-Thompson (also called Kac-Ray) inequality and a gamut of new inequalities for the spectral zeta function and other functions.

In this work, we use the Bethe sum rule and transform methods to prove a parallel set of inequalities for the eigenvalues of the fixed membrane problem. We also prove the equivalence of the Berezin-Li-Yau inequality with a classical inequality of Kac, analyze the work of A. Melas in a new light, and propose new conjectures.

This is joint work with Professor Evans Harrell of Georgia Tech.

A preprint is available at arxiv.org/abs/0712.4088