Self Adjoint Extension Schemes And Modern Applications To Quantum Hamiltonians


Self Adjoint Extension Schemes And Modern Applications To Quantum Hamiltonians
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Self Adjoint Extension Schemes And Modern Applications To Quantum Hamiltonians


Self Adjoint Extension Schemes And Modern Applications To Quantum Hamiltonians
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Author : Matteo Gallone
language : en
Publisher: Springer Nature
Release Date : 2023-04-04

Self Adjoint Extension Schemes And Modern Applications To Quantum Hamiltonians written by Matteo Gallone and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-04-04 with Science categories.


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.



Self Adjoint Extensions In Quantum Mechanics


Self Adjoint Extensions In Quantum Mechanics
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Author : D.M. Gitman
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-04-27

Self Adjoint Extensions In Quantum Mechanics written by D.M. Gitman and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-04-27 with Science categories.


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.



Applications Of Self Adjoint Extensions In Quantum Physics


Applications Of Self Adjoint Extensions In Quantum Physics
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Author : Pavel Exner
language : en
Publisher: Springer
Release Date : 2014-08-23

Applications Of Self Adjoint Extensions In Quantum Physics written by Pavel Exner and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-08-23 with Science categories.


The shared purpose in this collection of papers is to apply the theory of self-adjoint extensions of symmetry operators in various areas of physics. This allows the construction of exactly solvable models in quantum mechanics, quantum field theory, high energy physics, solid-state physics, microelectronics and other fields. The 20 papers selected for these proceedings give an overview of this field of research unparallelled in the published literature; in particular the views of the leading schools are clearly presented. The book will be an important source for researchers and graduate students in mathematical physics for many years to come. In these proceedings, researchers and graduate students in mathematical physics will find ways to construct exactly solvable models in quantum mechanics, quantum field theory, high energy physics, solid-state physics, microelectronics and other fields.



Non Selfadjoint Operators In Quantum Physics


Non Selfadjoint Operators In Quantum Physics
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Author : Fabio Bagarello
language : en
Publisher: John Wiley & Sons
Release Date : 2015-07-24

Non Selfadjoint Operators In Quantum Physics written by Fabio Bagarello and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-07-24 with Science categories.


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.



Non Hermitian Hamiltonians In Quantum Physics


Non Hermitian Hamiltonians In Quantum Physics
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Author : Fabio Bagarello
language : en
Publisher: Springer
Release Date : 2016-05-27

Non Hermitian Hamiltonians In Quantum Physics written by Fabio Bagarello and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-05-27 with Science categories.


This book presents the Proceedings of the 15th International Conference on Non-Hermitian Hamiltonians in Quantum Physics, held in Palermo, Italy, from 18 to 23 May 2015. Non-Hermitian operators, and non-Hermitian Hamiltonians in particular, have recently received considerable attention from both the mathematics and physics communities. There has been a growing interest in non-Hermitian Hamiltonians in quantum physics since the discovery that PT-symmetric Hamiltonians can have a real spectrum and thus a physical relevance. The main subjects considered in this book include: PT-symmetry in quantum physics, PT-optics, Spectral singularities and spectral techniques, Indefinite-metric theories, Open quantum systems, Krein space methods, and Biorthogonal systems and applications. The book also provides a summary of recent advances in pseudo-Hermitian Hamiltonians and PT-symmetric Hamiltonians, as well as their applications in quantum physics and in the theory of open quantum systems.



Introduction To Quantum Mechanics


Introduction To Quantum Mechanics
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Author : Horst R. Beyer
language : en
Publisher: Springer Nature
Release Date :

Introduction To Quantum Mechanics written by Horst R. Beyer and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on with categories.




Self Adjoint Extensions In Quantum Mechanics


Self Adjoint Extensions In Quantum Mechanics
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Author : Springer
language : en
Publisher:
Release Date : 2012-04-28

Self Adjoint Extensions In Quantum Mechanics written by Springer and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-04-28 with categories.




Mathematical Challenges Of Zero Range Physics


Mathematical Challenges Of Zero Range Physics
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Author : Alessandro Michelangeli
language : en
Publisher: Springer Nature
Release Date : 2021-02-04

Mathematical Challenges Of Zero Range Physics written by Alessandro Michelangeli and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-02-04 with Science categories.


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.



From Micro To Macro Quantum Systems


From Micro To Macro Quantum Systems
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Author : K Kong Wan
language : en
Publisher: World Scientific
Release Date : 2006-03-03

From Micro To Macro Quantum Systems written by K Kong Wan and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-03-03 with Science categories.


Traditional quantum theory has a very rigid structure, making it difficult to accommodate new properties emerging from novel systems. This book presents a flexible and unified theory for physical systems, from micro and macro quantum to classical. This is achieved by incorporating superselection rules and maximal symmetric operators into the theory. The resulting theory is applicable to classical, microscopic quantum and non-orthodox mixed quantum systems of which macroscopic quantum systems are examples. A unified formalism also greatly facilitates the discussion of interactions between these systems. A scheme of quantization by parts is introduced, based on the mathematics of selfadjoint and maximal symmetric extensions of symmetric operators, to describe point interactions. The results are applied to treat superconducting quantum circuits in various configurations. This book also discusses various topics of interest such as the asymptotic treatment of quantum state preparation and quantum measurement, local observables and local values, Schrödinger's cat states in superconducting systems, and a path space formulation of quantum mechanics. This self-contained book is complete with a review of relevant geometric and operator theories, for example, vector fields and operators, symmetric operators and their maximal symmetric extensions, direct integrals of Hilbert spaces and operators. Contents:Aspects of Geometric and Operator Theories:Manifolds and Dynamical SystemsOperators and Their Direct IntegralsOrthodox and Generalized Quantum Mechanics:Orthodox Quantum MechanicsPhysical Theory in Hilbert SpaceGeneralized Quantum MechanicsPoint Interactions, Macroscopic Quantum Systems and Superselection Rules:Point InteractionsMacroscopic Quantum SystemsAsymptotic Disjointness, Asymptotic Separability, Quantum Mechanics on Path Space and Superselection Rules:Separability and DecoherenceQuantum Mechanics on Path Space Readership: Theoretical and mathematical physicists, applied and pure mathematicians, physicists and philosophers of science (with an interest in quantum theory). Key Features:Rigorous formulation of a unified theory in a form directly applicable to physical systemsIntroduction of a quantization-by-part scheme to treat point interactionsSystematic and explicit treatment of quantum circuits in terms of point interactionsDistinctive selection of materials rarely discussed elsewhere, including a large number of examples and contemporary topicsDiscussions on the interplay of mathematics and physicsKeywords:Quantum Mechanics;Quantization;Macroscopic Quantum Systems;Superconducting Circuits;Point Contact InteractionsReviews:“Numerous sections of the book can be studied (and are really worth studying) like a textbook and without the necessity of going through the rest of the volume … certainly, everyone who works through the book will be rewarded by an enhanced comprehension of orthodox quantum theory … there are many solid reasons for recommending this book to the whole community of physicists and mathematicians — from graduate students to researchers — interested in a fresh description of the microscopic and macroscopic quantum worlds.”Mathematical Reviews



Many Body Quantum Theory In Condensed Matter Physics


Many Body Quantum Theory In Condensed Matter Physics
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Author : Henrik Bruus
language : en
Publisher: Oxford University Press
Release Date : 2004-09-02

Many Body Quantum Theory In Condensed Matter Physics written by Henrik Bruus and has been published by Oxford University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004-09-02 with Science categories.


The book is an introduction to quantum field theory applied to condensed matter physics. The topics cover modern applications in electron systems and electronic properties of mesoscopic systems and nanosystems. The textbook is developed for a graduate or advanced undergraduate course with exercises which aim at giving students the ability to confront real problems.