Quantum Mathematical Physics

This edition combines the earlier two volumes on Quantum Mechanics of Atoms and Molecules and on Quantum Mechanics of Large Systems, thus including in a single volume the material for a two-semester course on quantum physics. Since this volume is already quite heavy, I could not include many new results which show how lively the subject is. I just want to mention that inequality (IV:4. 1. l. 1) has been sharpened by T. Weidl by a factor 2][ and the difficult problem 1 of (III:4. 6) has been solved by A. Martin. I have to thank N. Ilieva for the devotion in preparing this new edition. Vienna, November 2001 Walter Thirring Preface to the Second Edition: Quantum Mechanics of Atoms and Molecules Ever since the first edition of this volume appeared in 1980 quantum statistical mechanics has florished. Innumerable results in many areas have been obtained and it would require a series of volumes to do justice to all of them. On the other hand the first edition was already rather crowded with many details so it would not be overburdened any more. Thus I added only one chapter on quantum ergodic theory where one can get the main notions across without too much pain. Nevertheless many subjects treated in the book had splendidely developed ever since and the only way out I could see is to add some recent references which the interested reader can consult.

This book presents different physical ideas and mathematical approaches in Quantum physics.

This is the first book to comprehensively cover quantum probabilistic approaches to spectral analysis of graphs, an approach developed by the authors. The book functions as a concise introduction to quantum probability from an algebraic aspect. Here readers will learn several powerful methods and techniques of wide applicability, recently developed under the name of quantum probability. The exercises at the end of each chapter help to deepen understanding.

Sergio Albeverio gave important contributions to many fields ranging from Physics to Mathematics, while creating new research areas from their interplay. Some of them are presented in this Volume that grew out of the Random Transformations and Invariance in Stochastic Dynamics Workshop held in Verona in 2019. To understand the theory of thermo- and fluid-dynamics, statistical mechanics, quantum mechanics and quantum field theory, Albeverio and his collaborators developed stochastic theories having strong interplays with operator theory and functional analysis. His contribution to the theory of (non Gaussian)-SPDEs, the related theory of (pseudo-)differential operators, and ergodic theory had several impacts to solve problems related, among other topics, to thermo- and fluid dynamics. His scientific works in the theory of interacting particles and its extension to configuration spaces lead, e.g., to the solution of open problems in statistical mechanics and quantum field theory. Together with Raphael Hoegh Krohn he introduced the theory of infinite dimensional Dirichlet forms, which nowadays is used in many different contexts, and new methods in the theory of Feynman path integration. He did not fear to further develop different methods in Mathematics, like, e.g., the theory of non-standard analysis and p-adic numbers.

This book presents the various types of coherent states introduced and studied in the physics and mathematics literature and describes their properties together with application to quantum physics problems. It is intended to serve as a compendium on coherent states and their applications for physicists and mathematicians, stretching from the basic mathematical structures of generalized coherent states in the sense of Perelomov via the semiclassical evolution of coherent states to various specific examples of coherent states (hydrogen atom, quantum oscillator, …).