Lieb-Thirring inequalities for higher order differential operators

TL;DR

This paper establishes Lieb-Thirring inequalities for higher order differential operators, including fourth order Schrödinger operators and polyharmonic operators, providing new bounds and insights into their spectral properties.

Contribution

It derives Lieb-Thirring inequalities for eigenvalues of higher order operators and proves a strict inequality case that challenges existing conjectures.

Findings

Lieb-Thirring inequalities derived for gamma >= 3/4

Extensions to polyharmonic operators and systems

Proved strict inequality in critical case for certain dimensions

Abstract

We derive Lieb-Thirring inequalities for the Riesz means of eigenvalues of order gamma >= 3/4 for fourth order Schr\"odinger operators in arbitrary dimensions. We also consider some extensions to polyharmonic operators, and to systems of such operators. For the critical case gamma = 1 - 1/2l in dimension d=1 with differential order 2l >= 4 we prove the strict inequality L^0(l,gamma,d) < L(l,gamma,d), which holds in contrast to current conjectures.