This work is dedicated to Pavel Exner on the occasion of his anniversary.\nThis volume is dedicated to Pavel Exner on the occasion of his 70th anniversary. It collects contributions by numerous scientists with expertise in mathematical physics and in particular in problems arising from quantum mechanics. The questions addressed in the contributions cover a large range of topics. A lot of attention was paid to differential operators with zero range interactions, which are often used as models in quantum mechanics. Several authors considered problems related to systems with mixed-dimensions such as quantum waveguides, quantum layers and quantum graphs. Eigenvalues and eigenfunctions of Laplace and Schrödinger operators are discussed too, as well as systems with adiabatic time evolution. Although most of the problems treated in the book have a quantum mechanical background, some contributions deal with issues which go well beyond this framework, for example the Cayley–Hamilton theorem, approximation formulae for contraction semigroups or factorization of analytic operator-valued Fredholm functions. As for the mathematical tools involved, the book provides a wide variety of techniques from functional analysis and operator theory.\nAltogether the volume presents a collection of research papers which will be of interest to any active scientist working in one of the above mentioned fields.
This work is dedicated to Pavel Exner on the occasion of his anniversary.\nThis volume is dedicated to Pavel Exner on the occasion of his 70th anniversary. It collects contributions by numerous scientists with expertise in mathematical physics and in particular in problems arising from quantum mechanics. The questions addressed in the contributions cover a large range of topics. A lot of attention was paid to differential operators with zero range interactions, which are often used as models in quantum mechanics. Several authors considered problems related to systems with mixed-dimensions such as quantum waveguides, quantum layers and quantum graphs. Eigenvalues and eigenfunctions of Laplace and Schrödinger operators are discussed too, as well as systems with adiabatic time evolution. Although most of the problems treated in the book have a quantum mechanical background, some contributions deal with issues which go well beyond this framework, for example the Cayley–Hamilton theorem, approximation formulae for contraction semigroups or factorization of analytic operator-valued Fredholm functions. As for the mathematical tools involved, the book provides a wide variety of techniques from functional analysis and operator theory.\nAltogether the volume presents a collection of research papers which will be of interest to any active scientist working in one of the above mentioned fields.
Dedicated to Pavel Exner on the occasion of his 70th birthday\nThis volume is dedicated to Pavel Exner on the occasion of his 70th anniversary. It collects contributions by numerous scientists with expertise in mathematical physics and in particular in problems arising from quantum mechanics. The questions addressed in the contributions cover a large range of topics. A lot of attention was paid to differential operators with zero range interactions, which are often used as models in quantum mechanics. Several authors considered problems related to systems with mixed-dimensions such as quantum waveguides, quantum layers and quantum graphs. Eigenvalues and eigenfunctions of Laplace and Schrödinger operators are discussed too, as well as systems with adiabatic time evolution. Although most of the problems treated in the book have a quantum mechanical background, some contributions deal with issues which go well beyond this framework, for example the Cayley–Hamilton theorem, approximation formulae for contraction semigroups or factorization of analytic operator-valued Fredholm functions. As for the mathematical tools involved, the book provides a wide variety of techniques from functional analysis and operator theory.\nAltogether the volume presents a collection of research papers which will be of interest to any active scientist working in one of the above mentioned fields.
Dedicated to Pavel Exner on the occasion of his 70th birthday\nThis volume is dedicated to Pavel Exner on the occasion of his 70th anniversary. It collects contributions by numerous scientists with expertise in mathematical physics and in particular in problems arising from quantum mechanics. The questions addressed in the contributions cover a large range of topics. A lot of attention was paid to differential operators with zero range interactions, which are often used as models in quantum mechanics. Several authors considered problems related to systems with mixed-dimensions such as quantum waveguides, quantum layers and quantum graphs. Eigenvalues and eigenfunctions of Laplace and Schrödinger operators are discussed too, as well as systems with adiabatic time evolution. Although most of the problems treated in the book have a quantum mechanical background, some contributions deal with issues which go well beyond this framework; for example the Cayley–Hamilton theorem, approximation formulae for contraction semigroups or factorization of analytic operator-valued Fredholm functions. As for the mathematical tools involved, the book provides a wide variety of techniques from functional analysis and operator theory.\nAltogether the volume presents a collection of research papers which will be of interest to any active scientist working in one of the above mentioned fields.
Quantum mechanics and the theory of operators on Hilbert space have been deeply linked since their beginnings in the early twentieth century. States of a quantum system correspond to certain elements of the configuration space and observables correspond to certain operators on the space. This book is a brief, but self-contained, introduction to the mathematical methods of quantum mechanics, with a view towards applications to Schrodinger operators. Part 1 of the book is a concise introduction to the spectral theory of unbounded operators. Only those topics that will be needed for later applications are covered. The spectral theorem is a central topic in this approach and is introduced at an early stage. Part 2 starts with the free Schrodinger equation and computes the free resolvent and time evolution. Position, momentum, and angular momentum are discussed via algebraic methods. Various mathematical methods are developed, which are then used to compute the spectrum of the hydrogen atom. Further topics include the nondegeneracy of the ground state, spectra of atoms, and scattering theory. This book serves as a self-contained introduction to spectral theory of unbounded operators in Hilbert space with full proofs and minimal prerequisites: Only a solid knowledge of advanced calculus and a one-semester introduction to complex analysis are required. In particular, no functional analysis and no Lebesgue integration theory are assumed. It develops the mathematical tools necessary to prove some key results in nonrelativistic quantum mechanics. Mathematical Methods in Quantum Mechanics is intended for beginning graduate students in both mathematics and physics and provides a solid foundation for reading more advanced books and current research literature. This new edition has additions and improvements throughout the book to make the presentation more student friendly.
A unique discussion of mathematical methods with applications to quantum mechanics Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects presents various mathematical constructions influenced by quantum mechanics and emphasizes the spectral theory of non-adjoint operators. Featuring coverage of functional analysis and algebraic methods in contemporary quantum physics, the book discusses recent emergence of the unboundedness of metric operators, which is a serious issue in the study of parity-time-symmetric quantum mechanics. The book also answers mathematical questions that are currently the subject of rigorous analysis, with potentially significant physical consequences. In addition to prompting a discussion of the role of mathematical methods in the contemporary development of quantum physics, the book features: * Chapter contributions written by well-known mathematical physicists who clarify numerous misunderstandings and misnomers while shedding light on new approaches in this growing area * An overview of recent inventions and advances in understanding functional analytic and algebraic methods for non-selfadjoint operators as well as the use of Krein space theory and pertubation theory * Rigorous support of the progress in theoretical physics of non-Hermitian systems in addition to mathematically justified applications in various domains of physics such as nuclear and particle physics, condensed matter physics An ideal reference, Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects is useful for researchers, professionals, and academics in applied mathematics and theoretical and/or applied physics who would like to expand their knowledge of classical applications of quantum tools to address problems in their research. Also a useful resource for recent and related trends, the book is appropriate as a graduate-level and/or PhD-level text for courses on quantum mechanics and mathematical models in physics.
This book is based on notes from the course developed and taught for more than 30 years at the Department of Mathematics of Leningrad University. The goal of the course was to present the basics of quantum mechanics and its mathematical content to students in mathematics. This book differs from the majority of other textbooks on the subject in that much more attention is paid to general principles of quantum mechanics. In particular, the authors describe in detail the relation between classical and quantum mechanics. When selecting particular topics, the authors emphasize those that are related to interesting mathematical theories. In particular, the book contains a discussion of problems related to group representation theory and to scattering theory. This book is rather elementary and concise, and it does not require prerequisites beyond the standard undergraduate mathematical curriculum. It is aimed at giving a mathematically oriented student the opportunity to grasp the main points of quantum theory in a mathematical framework.
This issue is devoted to our colleague and friend Pierre Duclos who passed away suddenly and prematurely in Prague on 12 January this year. We want to honour his memory in the way he would have liked, by collecting fresh and original work from his area of interest. Pierre Duclos was born on 8 January 1948 in Paris. He started as an engineering student but also attended graduate courses at the Centre de Physique Théorique (CPT) in Marseille, which inspired him to change his path and pursue the professional career of a researcher. He joined the Mathematical Physics team at CPT and obtained a position at the University of Toulon, where he later became a full professor. He was never solitary; always being full of energy and a smart and sociable person, he started and maintained many international collaborations and organized numerous conferences and seminars. In the early eighties he had strong ties with the Free and Technical Universities of Berlin. From the beginning of the nineties, he collaborated with coleagues in Prague, Bucharest, Santiago de Chile, and more recently also in Aalborg and Dublin. His scientific interests were wide, with a focus on mathematical methods of quantum theory. He made important contributions to our understanding of multiple-well Schrödinger operators, geometrically-induced properties of quantum waveguides, spectra of Wannier–Stark systems, dynamics with time-periodic perturbations, and transport in mesoscopic systems, to name his most significant results. We choose for this issue the title `Spectral and transport properties of quantum systems' which cover the subjects of most papers to which his colleagues, and often coauthors, contributed. We have also included a few other papers with topics related to Pierre's work. We are glad we were able to gather a numerous collection of papers which in our view represent interesting new developments. A few of them are works which bear Pierre's signature and have been completed by his collaborators. Moreover, we are aware of several other works which their authors dedicated to Pierre's memory but which for some reason or another did not make it to this issue. A list of these papers can be found at the end of the preface. This multitude of memorial papers shows that Pierre was popular not only as a colleague, coauthor, and teacher, but also as a person. Those who had the good fortune to work with him will always recall his blend of hard-working habits, strong views, and human warmth, which made him so unique. He will be remembered with gratitude and admiration by all who knew him well. We will miss him a lot. Jean-Michel Combes , Université du Sud Toulon-Var and Centre de Physique Théorique, CNRS Marseille, France Pavel Exner , Doppler Institute and Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of Sciences Valentin A Zagrebnov , Université de la Mediterranée and Centre de Physique Théorique, CNRS Marseille, France Guest Editors Other works dedicated to Pierre's memory: Aschbacher W, Barbaroux J-M, Faupin J and Guillot J-C 2010 Spectral theory for a mathematical model of weak interactions: the decay of the intermediate bosons W ± . II Annales Henri Poincaré at press Bellissard J and Palmer I 2009 The Jewett–Krieger construction for tilings arXiv:0906.2997 Gesztesy F and Zinchenko M 2010 Symmetrized perturbation determinants and applications to boundary data maps and Krein-type resolvent formulas arXiv:1007.4605 Kostrykin V, Potthoff J and Schrader R 2010 Brownian motions on metric graphs: Feller Brownian motions on intervals revisited arXiv:1008.3761 Stollmann P 2010 From uncertainty principles to Wegner estimates Math. Phys. Anal. Geom. 13 145–57