Spectral And Dynamical Properties Of Certain Quantum Hamiltonians In Dimension Two
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Spectral And Dynamical Properties Of Certain Quantum Hamiltonians In Dimension Two
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Author : Josef Mehringer
language : en
Publisher:
Release Date : 2015
Spectral And Dynamical Properties Of Certain Quantum Hamiltonians In Dimension Two written by Josef Mehringer and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015 with categories.
Open Quantum Systems I
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Author : Stéphane Attal
language : en
Publisher: Springer
Release Date : 2006-08-18
Open Quantum Systems I written by Stéphane Attal and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-08-18 with Mathematics categories.
Understanding dissipative dynamics of open quantum systems remains a challenge in mathematical physics. This problem is relevant in various areas of fundamental and applied physics. Significant progress in the understanding of such systems has been made recently. These books present the mathematical theories involved in the modeling of such phenomena. They describe physically relevant models, develop their mathematical analysis and derive their physical implications.
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.
Spectral Measures And Dynamics Typical Behaviors
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Author : Moacir Aloisio
language : en
Publisher: Springer Nature
Release Date : 2023-10-27
Spectral Measures And Dynamics Typical Behaviors written by Moacir Aloisio and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-10-27 with Science categories.
This book convenes and deepens generic results about spectral measures, many of them available so far in scattered literature. It starts with classic topics such as Wiener lemma, Strichartz inequality, and the basics of fractal dimensions of measures, progressing to more advanced material, some of them developed by the own authors. A fundamental concept to the mathematical theory of quantum mechanics, the spectral measure relates to the components of the quantum state concerning the energy levels of the Hamiltonian operator and, on the other hand, to the dynamics of such state. However, these correspondences are not immediate, with many nuances and subtleties discovered in recent years. A valuable example of such subtleties is found in the so-called “Wonderland theorem” first published by B. Simon in 1995. It shows that, for some metric space of self-adjoint operators, the set of operators whose spectral measures are singular continuous is a generic set (which, for some, is exotic). Recent works have revealed that, on top of singular continuity, there are other generic properties of spectral measures. These properties are usually associated with a number of different notions of generalized dimensions, upper and lower dimensions, with dynamical implications in quantum mechanics, ergodicity of dynamical systems, and evolution semigroups. All this opens ways to new and instigating avenues of research. Graduate students with a specific interest in the spectral properties of spectral measure are the primary target audience for this work, while researchers benefit from a selection of important results, many of them presented in the book format for the first time.
Proceedings Of The Second International Symposium On Quantum Theory And Symmetries
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Author : Andrzej Horzela
language : en
Publisher: World Scientific
Release Date : 2002
Proceedings Of The Second International Symposium On Quantum Theory And Symmetries written by Andrzej Horzela and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002 with Science categories.
This book presents the up-to-date status of quantum theory and the outlook for its development in the 21st century. The covered topics include basic problems of quantum physics, with emphasis on the foundations of quantum theory, quantum computing and control, quantum optics, coherent states and Wigner functions, as well as on methods of quantum physics based on Lie groups and algebras, quantum groups and noncommutative geometry.
Spectral Properties Of Multi Dimensional Quantum Spin Systems
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Author : Amanda M Young
language : en
Publisher:
Release Date : 2016
Spectral Properties Of Multi Dimensional Quantum Spin Systems written by Amanda M Young and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016 with categories.
In this dissertation, we analyze spectral properties of frustration-free quantum spin systems, with an emphasis on multi-dimensional models. The focus of this analysis is two-fold. First, we determine the existence or non-existence of a spectral gap above the ground state for a class of multi-dimensional quantum spin systems. Second, we determine a set of criteria for which the spectral gap above the ground state energy of a frustration-free quantum spin system is stable in the presence of local perturbations. We begin by providing the mathematical formulation of quantum spin systems, including a discussion of the thermodynamic limit. We formalize the construction of infinite volume systems when we introduce Lieb-Robinson bounds. When these bounds hold, the dynamics of the finite volume quantum spin systems extend to the thermodynamic limit, and a well-defined infinite volume Hamiltonian, known as the GNS Hamiltonian, is guaranteed to exist. Determining if a quantum spin system is gapped or gapless is a question of the spectrum of the GNS Hamiltonian. It is gapped if there is an interval above the ground state energy that does not contain any points of the spectrum. Otherwise, it is gapless. We show that to prove a gap for the GNS Hamiltonian of a frustration-free system, it suffices to obtain a uniform lower bound for the spectral gap in a sequence of finite volumes. To obtain these results we apply the martingale method, which was first introduced for frustration-free quantum spin systems with open boundary conditions by Nachtergaele in 1996 [48]. We extend this method to hold for periodic boundary conditions, and apply it to the class of pure finitely correlated state models with periodic boundary conditions.We then study the the one-species, multi-dimensional PVBS models and determine the exact collection of model parameters for which the models are gapped in the thermodynamic limit. The PVBS models are a class of particle hopping models where the model parameters dictate the direction a particle prefers to move. We consider these models on two types of infinite volumes. First, we take the thermodynamic limit of the PVBS model to the d-dimensional integer lattice. Second, we consider the PVBS model on a half space of the integer lattice where the boundary is defined by a hyperplane. We find two types of closures of the spectral gap. First, there is a bulk closure due to a critical change in the set of model parameters. The second type of closure is only found in the case of the hyperplane boundary, and is due to the creation of edge states across the boundary. This is inherently different from the bulk closure, as away from the boundary the model still behaves like a gapped system. We then transition to considering the stability of gapped ground state phases. Stability results establish that if a system is gapped, it will remain gapped under the addition of small perturbations. The two main technical tools we use to prove such a result are Lieb-Robinson bounds and the spectral flow. Common to both tools is that they express locality properties for observables under a global evolution. We generalize this notion by introducing the theory of quasi-local maps. We apply our theory to prove the stability result from [47], which holds for both one-dimensional andmulti-dimensional quantum spin systems. The result from [47], which we extend to hold for a larger class of models and boundary conditions, states that a uniformly gapped family of frustration-free Hamiltonians with topologically ordered ground states will remain gapped in the presence of an exponentially decaying perturbation. We conclude with applying the stability result the finitely correlated state models with periodic boundary conditions.
Dynamical Groups And Spectrum Generating Algebras In 2 Volumes
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Author : Arno Bohm
language : en
Publisher: World Scientific
Release Date : 1988-12-01
Dynamical Groups And Spectrum Generating Algebras In 2 Volumes written by Arno Bohm and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 1988-12-01 with Science categories.
This book contains comprehensive reviews and reprints on dynamical groups, spectrum generating algebras and spectrum supersymmetries, and their applications in atomic and molecular physics, nuclear physics, particle physics, and condensed matter physics. It is an important source for researchers as well as students who are doing courses on Quantum Mechanics and Advanced Quantum Mechanics.
Spectral Properties Of Hamiltonian Operators
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Author : K. Jorgens
language : en
Publisher:
Release Date : 2014-01-15
Spectral Properties Of Hamiltonian Operators written by K. Jorgens and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-01-15 with categories.
Spectral Properties Of Hamiltonian Operators
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Author : K. Jörgens
language : en
Publisher: Springer
Release Date : 1973-04-20
Spectral Properties Of Hamiltonian Operators written by K. Jörgens and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 1973-04-20 with Mathematics categories.
Spectral Theory Of Schrodinger Operators
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Author : Rafael del Río
language : en
Publisher: American Mathematical Soc.
Release Date : 2004
Spectral Theory Of Schrodinger Operators written by Rafael del Río and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004 with Mathematics categories.
This volume gathers the articles based on a series of lectures from a workshop held at the Institute of Applied Mathematics of the National University of Mexico. The aim of the book is to present to a non-specialized audience the basic tools needed to understand and appreciate new trends of research on Schrodinger operator theory. Topics discussed include various aspects of the spectral theory of differential operators, the theory of self-adjoint operators, finite rank perturbations, spectral properties of random Schrodinger operators, and scattering theory for Schrodinger operators. The material is suitable for graduate students and research mathematicians interested in differential operators, in particular, spectral theory of Schrodinger operators.