Inequalities play a fundamental role in Functional Analysis and it is widely recognized that finding them, especially sharp estimates, is an art. E. H. Lieb has discovered a host of inequalities that are enormously useful in mathematics as well as in physics. His results are collected in this book which should become a standard source for further research. Together with the mathematical proofs the author also presents numerous applications to the calculus of variations and to many problems of quantum physics, in particular to atomic physics.

This is the third Selecta of publications of Elliott Lieb, the first two being Stabil ity of Matter: From Atoms to Stars, edited by Walter Thirring, and Inequalities, edited by Michael Loss and Mary Beth Ruskai. A companion fourth Selecta on Statistical Mechanics is also edited by us. Elliott Lieb has been a pioneer of the discipline of mathematical physics as it is nowadays understood and continues to lead several of its most active directions today. For the first part of this selecta we have made a selection of Lieb's works on Condensed Matter Physics. The impact of Lieb's work in mathematical con densed matter physics is unrivaled. It is fair to say that if one were to name a founding father of the field, Elliott Lieb would be the only candidate to claim this singular position. While in related fields, such as Statistical Mechanics and Atomic Physics, many key problems are readily formulated in unambiguous mathematical form, this is less so in Condensed Matter Physics, where some say that rigor is "probably impossible and certainly unnecessary." By carefully select ing the most important questions and formulating them as well-defined mathemat ical problems, and then solving a good number of them, Lieb has demonstrated the quoted opinion to be erroneous on both counts. What is true, however, is that many of these problems turn out to be very hard. It is not unusual that they take a decade (even several decades) to solve."

This book contains a unique survey of the mathematically rigorous results about the quantum-mechanical many-body problem that have been obtained by the authors in the past seven years. It is a topic that is not only rich mathematically, using a large variety of techniques in mathematical analysis, but it is also one with strong ties to current experiments on ultra-cold Bose gases and Bose-Einstein condensation. It is an active subject of ongoing research, and this book provides a pedagogical entry into the field for graduate students and researchers. It is an outgrowth of a course given by the authors for graduate students and post-doctoral researchers at the Oberwolfach Research Institute in 2004. The book also provides a coherent summary of the field and a reference for mathematicians and physicists active in research on quantum mechanics.

This is the third Selecta of publications of Elliott Lieb, the first two being Stabil ity of Matter: From Atoms to Stars, edited by Walter Thirring, and Inequalities, edited by Michael Loss and Mary Beth Ruskai. A companion fourth Selecta on Statistical Mechanics is also edited by us. Elliott Lieb has been a pioneer of the discipline of mathematical physics as it is nowadays understood and continues to lead several of its most active directions today. For the first part of this selecta we have made a selection of Lieb's works on Condensed Matter Physics. The impact of Lieb's work in mathematical con densed matter physics is unrivaled. It is fair to say that if one were to name a founding father of the field, Elliott Lieb would be the only candidate to claim this singular position. While in related fields, such as Statistical Mechanics and Atomic Physics, many key problems are readily formulated in unambiguous mathematical form, this is less so in Condensed Matter Physics, where some say that rigor is "probably impossible and certainly unnecessary." By carefully select ing the most important questions and formulating them as well-defined mathemat ical problems, and then solving a good number of them, Lieb has demonstrated the quoted opinion to be erroneous on both counts. What is true, however, is that many of these problems turn out to be very hard. It is not unusual that they take a decade (even several decades) to solve."

This fourth edition of selecta of my work on the stability of matter contains recent work on two topics that continue to fascinate me: Quantum electrodynamics (QED) and the Bose gas. Three papers have been added to Part VII on QED. As I mentioned in the preface to the third edition, there must be a way to formulate a non-perturbative QED, presumably with an ultraviolet cutoff, that correctly describes low energy physics, i.e., ordinary matter and its interaction with the electromagnetic field. The new paper VII.5, which quantizes the results in V.9, shows that the elementary no-pair version of relativistic QED (using the Dirac operator) is unstable when many-body effects are taken into account. Stability can be restored, however, if the Dirac operator with the field, instead of the bare Dirac operator, is used to define an electron. Thus, the notion of a bare electron without its self-field is physically questionable."

Inequalities play a fundamental role in Functional Analysis and it is widely recognized that finding them, especially sharp estimates, is an art. E. H. Lieb has discovered a host of inequalities that are enormously useful in mathematics as well as in physics. His results are collected in this book which should become a standard source for further research. Together with the mathematical proofs the author also presents numerous applications to the calculus of variations and to many problems of quantum physics, in particular to atomic physics.

Significantly revised and expanded, this second edition provides readers at all levels - from beginning students to practising analysts - with the basic concepts and standard tools necessary to solve problems of analysis, and how to apply these concepts to research in a variety of areas. The authors quickly move from basic topics, to methods that work successfully in mathematics and its applications. While omitting many usual typical textbook topics, this volume includes all necessary definitions, proofs, explanations, examples, and exercises to bring the reader to an advanced level of understanding with a minimum of fuss, and, at the same time, doing so in a rigorous and pedagogical way. Many topics that are useful and important, but usually left to advanced monographs, are presented, and these should give the beginner a sense that the subject is alive and growing.

Some of the articles in this collection give up-to-date accounts of areas in mathematical physics to which Valentine Bargmann made pioneering contributions. The others treat a selection of the most interesting current topics in the field. The contributions include both reviews and original results. Contents: The Inverse r-Squared Force (Henry D. I. Abarbanel; Certain Hilbert Spaces of Analytic Functions Associated with the Heisenberg Group (Donald Babbitt); Lower Bound for the Ground State Energy of the Schrodinger Equation Using the Sharp Form of Young's Inequality (John F. Barnes, Herm Jan Brascamp, and Elliott II. Lieb); Alternative Theories of Gravitation (Peter G. Bergmann; )Generalized Wronskian Relations (F. Calogero); Old and New Approaches to the Inverse-Scattering Problem (Freeman J. Dyson); A Family of Optimal Conditions for the Absence of Bound States in a Potential (V. Glaser, A. Martin, H. Grosse, and W. Thirring); Spinning Tops in External Fields (Sergio Hojman and Tullio Regge); Measures on the Finite Dimensional Subspaces of a Hilbert Space (Res Jost); The Froissart Bound and Crossing Symmetry (N. N. Khuri); Intertwining Operators for SL(n,R) (A. W. Knapp and E. M. Stein); Inequalities for the Moments of the Eigenvalues of the Schrodinger Hamiltonian and Their Relations to Sobolev Inequalities (Elliott H. Lieb and Walter Thirriny); On the Number of Bound States of Two Body Schrodinger Operators (Barry Simon); Quantum Dynamics: From Automorphism to Hamiltonian (Barry Simon); Semiclassical Analysis Illuminates the Connection between Potential and Bound States and Scattering (John Archibald Wheeler); Instability Phenomena in the External Field Problem for Two Classes of Relativistic Wave Equations (A. S. Wightman) Originally published in 1976. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Some of the articles in this collection give up-to-date accounts of areas in mathematical physics to which Valentine Bargmann made pioneering contributions. The others treat a selection of the most interesting current topics in the field. The contributions include both reviews and original results. Contents: The Inverse r-Squared Force (Henry D. I. Abarbanel; Certain Hilbert Spaces of Analytic Functions Associated with the Heisenberg Group (Donald Babbitt); Lower Bound for the Ground State Energy of the Schrodinger Equation Using the Sharp Form of Young's Inequality (John F. Barnes, Herm Jan Brascamp, and Elliott II. Lieb); Alternative Theories of Gravitation (Peter G. Bergmann; )Generalized Wronskian Relations (F. Calogero); Old and New Approaches to the Inverse-Scattering Problem (Freeman J. Dyson); A Family of Optimal Conditions for the Absence of Bound States in a Potential (V. Glaser, A. Martin, H. Grosse, and W. Thirring); Spinning Tops in External Fields (Sergio Hojman and Tullio Regge); Measures on the Finite Dimensional Subspaces of a Hilbert Space (Res Jost); The Froissart Bound and Crossing Symmetry (N. N. Khuri); Intertwining Operators for SL(n,R) (A. W. Knapp and E. M. Stein); Inequalities for the Moments of the Eigenvalues of the Schrodinger Hamiltonian and Their Relations to Sobolev Inequalities (Elliott H. Lieb and Walter Thirriny); On the Number of Bound States of Two Body Schrodinger Operators (Barry Simon); Quantum Dynamics: From Automorphism to Hamiltonian (Barry Simon); Semiclassical Analysis Illuminates the Connection between Potential and Bound States and Scattering (John Archibald Wheeler); Instability Phenomena in the External Field Problem for Two Classes of Relativistic Wave Equations (A. S. Wightman) Originally published in 1976. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.