A positive density analogue of the Lieb-Thirring inequality

TL;DR

This paper extends Lieb-Thirring inequalities to positive density scenarios, providing bounds on perturbations of the Fermi sea in quantum systems, which is relevant for understanding spectral properties of many-particle systems.

Contribution

It introduces a novel positive density analogue of the Lieb-Thirring inequality for perturbations of the Fermi sea, expanding the scope of spectral bounds in quantum mechanics.

Findings

Derived new inequalities for the continuous spectrum of the Laplacian.

Established bounds for perturbations of the Fermi sea.

Extended the applicability of Lieb-Thirring inequalities to positive density contexts.

Abstract

The Lieb-Thirring inequalities give a bound on the negative eigenvalues of a Schr\"odinger operator in terms of an $L^p$ norm of the potential. This is dual to a bound on the $H^1$-norms of a system of orthonormal functions. Here we extend these to analogous inequalities for perturbations of the Fermi sea of non-interacting particles, i.e., for perturbations of the continuous spectrum of the Laplacian by local potentials.