The state of the Lieb–Thirring conjecture

Abstract

In 1976 Lieb and Thirring established upper bounds on sums of powers of the negative eigenvalues of a Schrödinger 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.Dedicated to Elliott H. Lieb on the occasion of his 90th birthday with gratitude for his mentorship and for all that he has taught me.