The state of the Lieb--Thirring conjecture

TL;DR

This paper reviews the progress over 45 years on the Lieb--Thirring conjecture, which involves optimal constants in inequalities related to eigenvalues of Schrödinger operators, highlighting key developments and current status.

Contribution

It provides a comprehensive review of the historical and recent advances concerning the Lieb--Thirring conjecture and its optimal constants.

Findings

Summary of known bounds and conjectured values for constants

Overview of methods used in recent investigations

Identification of open problems and future directions

Abstract

In 1976 Lieb and Thirring established upper bounds on sums of powers of the negative eigenvalues of a Schr\"odinger operator in terms of semiclassical phase-space integrals. Over the last 45 years the optimal constants in these inequalities, the values of which were conjectured by Lieb and Thirring, have been subject of intense investigations. We aim to review existing results.