Lieb–Thirring inequalities for higher order differential operators

Abstract

We derive Lieb-Thirring inequalities for the Riesz means of eigenvalues of order gamma >= 3/4 for a fourth order operator in arbitrary dimensions. We also consider some extensions to polyharmonic operators, and to systems of such operators, in dimensions greater than one. For the critical case gamma=1-1/(2l) in dimension d=1 with l >= 2 we prove the inequality L-l,r,d(o) < L-l,L-r,L-d, which holds in contrast to current conjectures.