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The analysis of eigenvalues of Laplace and Schrödinger operators is an important and classical topic in mathematical physics with many applications. This book presents a thorough introduction to the area, suitable for masters and graduate students, and includes an ample amount of background material on the spectral theory of linear operators in Hilbert spaces and on Sobolev space theory. Of particular interest is a family of inequalities by Lieb and Thirring on eigenvalues of Schrödinger operators, which they used in their proof of stability of matter. The final part of this book is devoted to the active research on sharp constants in these inequalities and contains state-of-the-art results, serving as a reference for experts and as a starting point for further research.
List of contents
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.
About the author
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.
Summary
Eigenvalues of Laplace and Schrödinger operators play a fundamental role in many applications in mathematics and physics. This graduate-level book is devoted to their qualitative and quantitative mathematical analysis. It assumes no prior knowledge in this area and leads up to cutting-edge research on sharp constants in Lieb–Thirring inequalities.
