Semiclassical bounds for spectra of biharmonic operators

Abstract

We provide complementary semiclassical bounds for the Riesz means R1(z) of the eigenvalues of various biharmonic operators, with a second term in the expected power of z. The method we discuss makes use of the averaged variational principle (AVP), and yields two-sided bounds for individual eigenvalues, which are semiclassically sharp. The AVP also yields comparisons with Riesz means of different operators, in particular Laplacians.

Keywords

• biharmonic operator
• riesz means
• eigenvalue asymptotics
• semiclassical bounds for eigenvalues
• averaged variational principle