Rupert L. Frank holds a chair in applied mathematics at LMU Munich and is doing research primarily in analysis and mathematical physics. He is an invited speaker at the 2022 International Congress of Mathematics. Ari Laptev is Professor at Imperial College London. His research interests include different aspects of spectral theory and functional inequalities. He is a member of the Royal Swedish Academy of Science, a Fellow of EurASc and a member of Academia Europaea. Timo Weidl is Professor at the University of Stuttgart. He works on spectral theory and mathematical physics.

• Overview; Part I. Background Material: 1. Elements of operator theory; 2. Elements of Sobolev space theory; Part II. The Laplace and Schrödinger Operators: 3. The Laplacian on a domain; 4. The Schrödinger operator; Part III. Sharp Constants in Lieb–Thirring Inequalities:5. Sharp Lieb–Thirring inequalities; 6. Sharp Lieb–Thirring inequalities in higher dimensions; 7. More on sharp Lieb–Thirring inequalities; 8. More on the Lieb–Thirring constants; References; Index.

'In 1975, Lieb and Thirring proved a remarkable bound of the sum of the negative eigenvalues of a Schrödinger operator in three dimensions in terms of the L^{5/2}-norm of the potential and used it in their proof of the stability of matter. Shortly thereafter, they realized it was a case of a lovely set of inequalities which generalize Sobolev inequalities and have come to be called Lieb-Thirring bounds. This has spawned an industry with literally hundreds of papers on extensions, generalizations and optimal constants. It is wonderful to have the literature presented and synthesized by three experts who begin by giving the background necessary for this book to be useful not only to specialists but to the novice wishing to understand a deep chapter in mathematical analysis.' Barry Simon, California Institute of Technology