Applications Of Self Adjoint Extensions In Quantum Physics

Author : D.M. Gitman,I.V. Tyutin,B.L. Voronov
Publisher : Springer Science & Business Media
Page : 523 pages
File Size : 53,8 Mb
Release : 2012-04-27
Category : Science
ISBN : 9780817646622

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Self-adjoint Extensions in Quantum Mechanics by D.M. Gitman,I.V. Tyutin,B.L. Voronov Pdf

This exposition is devoted to a consistent treatment of quantization problems, based on appealing to some nontrivial items of functional analysis concerning the theory of linear operators in Hilbert spaces. The authors begin by considering quantization problems in general, emphasizing the nontriviality of consistent operator construction by presenting paradoxes to the naive treatment. It then builds the necessary mathematical background following it by the theory of self-adjoint extensions. By considering several problems such as the one-dimensional Calogero problem, the Aharonov-Bohm problem, the problem of delta-like potentials and relativistic Coulomb problemIt then shows how quantization problems associated with correct definition of observables can be treated consistently for comparatively simple quantum-mechanical systems. In the end, related problems in quantum field theory are briefly introduced. This well-organized text is most suitable for students and post graduates interested in deepening their understanding of mathematical problems in quantum mechanics. However, scientists in mathematical and theoretical physics and mathematicians will also find it useful.

Author : Pavel Exner,Petr Seba
Publisher : Unknown
Page : 288 pages
File Size : 52,7 Mb
Release : 2014-01-15
Category : Electronic
ISBN : 3662137615

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Applications of Self-Adjoint Extensions in Quantum Physics by Pavel Exner,Petr Seba Pdf

Author : Matteo Gallone,Alessandro Michelangeli
Publisher : Springer Nature
Page : 557 pages
File Size : 51,9 Mb
Release : 2023-04-04
Category : Science
ISBN : 9783031108853

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Self-Adjoint Extension Schemes and Modern Applications to Quantum Hamiltonians by Matteo Gallone,Alessandro Michelangeli Pdf

This book introduces and discusses the self-adjoint extension problem for symmetric operators on Hilbert space. It presents the classical von Neumann and Krein–Vishik–Birman extension schemes both in their modern form and from a historical perspective, and provides a detailed analysis of a range of applications beyond the standard pedagogical examples (the latter are indexed in a final appendix for the reader’s convenience). Self-adjointness of operators on Hilbert space representing quantum observables, in particular quantum Hamiltonians, is required to ensure real-valued energy levels, unitary evolution and, more generally, a self-consistent theory. Physical heuristics often produce candidate Hamiltonians that are only symmetric: their extension to suitably larger domains of self-adjointness, when possible, amounts to declaring additional physical states the operator must act on in order to have a consistent physics, and distinct self-adjoint extensions describe different physics. Realising observables self-adjointly is the first fundamental problem of quantum-mechanical modelling. The discussed applications concern models of topical relevance in modern mathematical physics currently receiving new or renewed interest, in particular from the point of view of classifying self-adjoint realisations of certain Hamiltonians and studying their spectral and scattering properties. The analysis also addresses intermediate technical questions such as characterising the corresponding operator closures and adjoints. Applications include hydrogenoid Hamiltonians, Dirac–Coulomb Hamiltonians, models of geometric quantum confinement and transmission on degenerate Riemannian manifolds of Grushin type, and models of few-body quantum particles with zero-range interaction. Graduate students and non-expert readers will benefit from a preliminary mathematical chapter collecting all the necessary pre-requisites on symmetric and self-adjoint operators on Hilbert space (including the spectral theorem), and from a further appendix presenting the emergence from physical principles of the requirement of self-adjointness for observables in quantum mechanics.

Author : Fabio Bagarello,Jean-Pierre Gazeau,Franciszek Hugon Szafraniec,Miloslav Znojil
Publisher : John Wiley & Sons
Page : 432 pages
File Size : 55,5 Mb
Release : 2015-07-24
Category : Science
ISBN : 9781118855263

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Non-Selfadjoint Operators in Quantum Physics by Fabio Bagarello,Jean-Pierre Gazeau,Franciszek Hugon Szafraniec,Miloslav Znojil Pdf

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 the recent emergence of 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 on 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 perturbation 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 and 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.

Author : Jirí Blank,Pavel Exner,Miloslav Havlícek
Publisher : Springer Science & Business Media
Page : 677 pages
File Size : 42,8 Mb
Release : 2008-09-24
Category : Science
ISBN : 9781402088704

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Hilbert Space Operators in Quantum Physics by Jirí Blank,Pavel Exner,Miloslav Havlícek Pdf

The new edition of this book detailing the theory of linear-Hilbert space operators and their use in quantum physics contains two new chapters devoted to properties of quantum waveguides and quantum graphs. The bibliography contains 130 new items.

Author : Springer
Publisher : Unknown
Page : 528 pages
File Size : 43,5 Mb
Release : 2012-04-28
Category : Electronic
ISBN : 0817671889

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Self-Adjoint Extensions in Quantum Mechanics by Springer Pdf

Author : César R. de Oliveira
Publisher : Springer Science & Business Media
Page : 410 pages
File Size : 48,7 Mb
Release : 2008-12-30
Category : Science
ISBN : 9783764387952

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Intermediate Spectral Theory and Quantum Dynamics by César R. de Oliveira Pdf

The spectral theory of linear operators plays a key role in the mathematical formulation of quantum theory. This textbook provides a concise and comprehensible introduction to the spectral theory of (unbounded) self-adjoint operators and its application in quantum dynamics. Many examples and exercises are included that focus on quantum mechanics.

Author : Horst R. Beyer
Publisher : Springer Nature
Page : 222 pages
File Size : 49,6 Mb
Release : 2024-06-30
Category : Electronic
ISBN : 9783031490781

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Introduction to Quantum Mechanics by Horst R. Beyer Pdf

Author : Gerald Teschl
Publisher : American Mathematical Soc.
Page : 322 pages
File Size : 52,8 Mb
Release : 2009
Category : Mathematics
ISBN : 9780821846605

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Mathematical Methods in Quantum Mechanics by Gerald Teschl Pdf

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. It is well suited for self-study and includes numerous exercises (many with hints).

Author : Pavel Kurasov,Ari Laptev,Sergey Naboko,Barry Simon
Publisher : Springer Nature
Page : 627 pages
File Size : 49,5 Mb
Release : 2020-07-14
Category : Mathematics
ISBN : 9783030315313

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Analysis as a Tool in Mathematical Physics by Pavel Kurasov,Ari Laptev,Sergey Naboko,Barry Simon Pdf

Boris Pavlov (1936-2016), to whom this volume is dedicated, was a prominent specialist in analysis, operator theory, and mathematical physics. As one of the most influential members of the St. Petersburg Mathematical School, he was one of the founders of the Leningrad School of Non-self-adjoint Operators. This volume collects research papers originating from two conferences that were organized in memory of Boris Pavlov: “Spectral Theory and Applications”, held in Stockholm, Sweden, in March 2016, and “Operator Theory, Analysis and Mathematical Physics – OTAMP2016” held at the Euler Institute in St. Petersburg, Russia, in August 2016. The volume also includes water-color paintings by Boris Pavlov, some personal photographs, as well as tributes from friends and colleagues.

Author : Manuel Gadella,Luiz A. Manzoni,José Tadeu Lunardi
Publisher : Frontiers Media SA
Page : 182 pages
File Size : 44,7 Mb
Release : 2021-03-12
Category : Science
ISBN : 9782889665921

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Contact Interactions in Quantum Mechanics: Theory, Mathematical Aspects and Applications by Manuel Gadella,Luiz A. Manzoni,José Tadeu Lunardi Pdf

Author : Alessandro Michelangeli
Publisher : Springer Nature
Page : 331 pages
File Size : 45,6 Mb
Release : 2021-02-04
Category : Science
ISBN : 9783030604530

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Mathematical Challenges of Zero-Range Physics by Alessandro Michelangeli Pdf

Since long over the decades there has been a large transversal community of mathematicians grappling with the sophisticated challenges of the rigorous modelling and the spectral and scattering analysis of quantum systems of particles subject to an interaction so much localised to be considered with zero range. Such a community is experiencing fruitful and inspiring exchanges with experimental and theoretical physicists. This volume reflects such spirit, with a diverse range of original contributions by experts, presenting an up-to-date collection of most relevant results and challenging open problems. It has been conceived with the deliberate two-fold purpose of serving as an updated reference for recent results, mathematical tools, and the vast related literature on the one hand, and as a bridge towards several key open problems that will surely form the forthcoming research agenda in this field.

Author : Malcolm Brown,Fritz Gesztesy,Pavel Kurasov,Ari Laptev,Barry Simon,Gunter Stolz,Ian Wood
Publisher : Springer Nature
Page : 731 pages
File Size : 55,7 Mb
Release : 2023-09-21
Category : Mathematics
ISBN : 9783031311390

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From Complex Analysis to Operator Theory: A Panorama by Malcolm Brown,Fritz Gesztesy,Pavel Kurasov,Ari Laptev,Barry Simon,Gunter Stolz,Ian Wood Pdf

This volume is dedicated to the memory of Sergey Naboko (1950-2020). In addition to original research contributions covering the vast areas of interest of Sergey Naboko, it includes personal reminiscences and comments on the works and legacy of Sergey Naboko’s scientific achievements. Areas from complex analysis to operator theory, especially, spectral theory, are covered, and the papers will inspire current and future researchers in these areas.

Author : Michael Reed,Barry Simon
Publisher : Elsevier
Page : 361 pages
File Size : 42,5 Mb
Release : 1975-11-05
Category : Mathematics
ISBN : 9780080925370

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II: Fourier Analysis, Self-Adjointness by Michael Reed,Barry Simon Pdf

This volume will serve several purposes: to provide an introduction for graduate students not previously acquainted with the material, to serve as a reference for mathematical physicists already working in the field, and to provide an introduction to various advanced topics which are difficult to understand in the literature. Not all the techniques and application are treated in the same depth. In general, we give a very thorough discussion of the mathematical techniques and applications in quatum mechanics, but provide only an introduction to the problems arising in quantum field theory, classical mechanics, and partial differential equations. Finally, some of the material developed in this volume will not find applications until Volume III. For all these reasons, this volume contains a great variety of subject matter. To help the reader select which material is important for him, we have provided a "Reader's Guide" at the end of each chapter.

Author : S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, H. Holden, and an appendix by P. Exner
Publisher : American Mathematical Soc.
Page : 508 pages
File Size : 55,5 Mb
Release : 2024-06-30
Category : Quantum theory
ISBN : 082186940X

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Solvable Models in Quantum Mechanics by S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, H. Holden, and an appendix by P. Exner Pdf

"This monograph presents a detailed study of a class of solvable models in quantum mechanics that describe the motion of a particle in a potential having support at the positions of a discrete (finite or infinite) set of point sources. Both situations–where the strengths of the sources and their locations are precisely known and where these are only known with a given probability distribution–are covered. The authors present a systematic mathematical approach to these models and illustrate its connections with previous heuristic derivations and computations. Results obtained by different methods in disparate contexts are thus unified and a systematic control over approximations to the models, in which the point interactions are replaced by more regular ones, is provided. The first edition of this book generated considerable interest for those learning advanced mathematical topics in quantum mechanics, especially those connected to the Schrödinger equations. This second edition includes a new appendix by Pavel Exner, who has prepared a summary of the progress made in the field since 1988. His summary, centering around two-body point interaction problems, is followed by a bibliography focusing on essential developments made since 1988. appendix by Pavel Exner, who has prepared a summary of the progress made in the field since 1988. His summary, centering around two-body point interaction problems, is followed by a bibliography focusing on essential developments made since 1988."--Résumé de l'éditeur.