Hardy-Lieb-Thirring inequalities for fractional Schroedinger operators

TL;DR

This paper extends Lieb-Thirring inequalities to fractional and relativistic Schrödinger operators with Hardy weights and magnetic fields, establishing new stability results for relativistic matter at critical nuclear charge.

Contribution

It proves Hardy-Lieb-Thirring inequalities for fractional and relativistic Schrödinger operators, including magnetic fields, and applies these to stability of relativistic matter at critical charge.

Findings

Lieb-Thirring inequalities hold with Hardy weights for fractional operators.

Established Sobolev inequalities for these operators.

Extended stability of relativistic matter up to critical nuclear charge.

Abstract

We show that the Lieb-Thirring inequalities on moments of negative eigenvalues of Schroedinger-like operators remain true, with possibly different constants, when the critical Hardy-weight $C|x|^{-2}$ is subtracted from the Laplace operator. We do so by first establishing a Sobolev inequality for such operators. Similar results are true for fractional powers of the Laplacian and the Hardy-weight and, in particular, for relativistic Schroedinger operators. We also allow for the inclusion of magnetic vector potentials. As an application, we extend, for the first time, the proof of stability of relativistic matter with magnetic fields all the way up to the critical value of the nuclear charge $Z\alpha=2/\pi$.