Equivalence of Sobolev inequalities and Lieb-Thirring inequalities

TL;DR

This paper demonstrates that Lieb-Thirring inequalities, which bound sums of eigenvalues of certain operators, can be derived from Sobolev inequalities for a broad class of kinetic energy operators, establishing a fundamental equivalence.

Contribution

It establishes a general equivalence between Sobolev and Lieb-Thirring inequalities for various kinetic energy operators, broadening the understanding of their interrelation.

Findings

Lieb-Thirring inequalities follow from Sobolev inequalities for general kinetic operators.

The results apply under very broad definitions of the kinetic energy operator.

This equivalence simplifies the derivation of spectral bounds in quantum mechanics.

Abstract

We show that, under very general definitions of a kinetic energy operator $T$, the Lieb-Thirring inequalities for sums of eigenvalues of $T-V$ can be derived from the Sobolev inequality appropriate to that choice of $T$.