Spectral Theory And Nonlinear Functional Analysis
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Spectral Theory And Nonlinear Functional Analysis
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Author : Julian Lopez-Gomez
language : en
Publisher: CRC Press
Release Date : 2001-03-28
Spectral Theory And Nonlinear Functional Analysis written by Julian Lopez-Gomez and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001-03-28 with Mathematics categories.
This Research Note addresses several pivotal problems in spectral theory and nonlinear functional analysis in connection with the analysis of the structure set of zeroes of a general class of nonlinear operators. Appealing to a broad audience, it contains many important contributions to linear algebra, linear functional analysis, nonlinear functional analysis, and topology. The author gives several applications of the abstract theory to reaction diffusion equations and systems. The results presented cover a thirty-year period and cut across a variety of mathematical fields.
Nonlinear Spectral Theory
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Author : Jürgen Appell
language : en
Publisher: Walter de Gruyter
Release Date : 2008-08-22
Nonlinear Spectral Theory written by Jürgen Appell and has been published by Walter de Gruyter this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-08-22 with Mathematics categories.
In view of the eminent importance of spectral theory of linear operators in many fields of mathematics and physics, it is not surprising that various attempts have been made to define and study spectra also for nonlinear operators. This book provides a comprehensive and self-contained treatment of the theory, methods, and applications of nonlinear spectral theory. The first chapter briefly recalls the definition and properties of the spectrum and several subspectra for bounded linear operators. Then some numerical characteristics for nonlinear operators are introduced which are useful for describing those classes of operators for which there exists a spectral theory. Since spectral values are closely related to solvability results for operator equations, various conditions for the local or global invertibility of a nonlinear operator are collected in the third chapter. The following two chapters are concerned with spectra for certain classes of continuous, Lipschitz continuous, or differentiable operators. These spectra, however, simply adapt the corresponding definitions from the linear theory which somehow restricts their applicability. Other spectra which are defined in a completely different way, but seem to have useful applications, are defined and studied in the following four chapters. The remaining three chapters are more application-oriented and deal with nonlinear eigenvalue problems, numerical ranges, and selected applications to nonlinear problems. The only prerequisite for understanding this book is a modest background in functional analysis and operator theory. It is addressed to non-specialists who want to get an idea of the development of spectral theory for nonlinear operators in the last 30 years, as well as a glimpse of the diversity of the directions in which current research is moving.
A Guide To Spectral Theory
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Author : Christophe Cheverry
language : en
Publisher: Springer Nature
Release Date : 2021-05-06
A Guide To Spectral Theory written by Christophe Cheverry and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-05-06 with Mathematics categories.
This textbook provides a graduate-level introduction to the spectral theory of linear operators on Banach and Hilbert spaces, guiding readers through key components of spectral theory and its applications in quantum physics. Based on their extensive teaching experience, the authors present topics in a progressive manner so that each chapter builds on the ones preceding. Researchers and students alike will also appreciate the exploration of more advanced applications and research perspectives presented near the end of the book. Beginning with a brief introduction to the relationship between spectral theory and quantum physics, the authors go on to explore unbounded operators, analyzing closed, adjoint, and self-adjoint operators. Next, the spectrum of a closed operator is defined and the fundamental properties of Fredholm operators are introduced. The authors then develop the Grushin method to execute the spectral analysis of compact operators. The chapters that follow are devoted to examining Hille-Yoshida and Stone theorems, the spectral analysis of self-adjoint operators, and trace-class and Hilbert-Schmidt operators. The final chapter opens the discussion to several selected applications. Throughout this textbook, detailed proofs are given, and the statements are illustrated by a number of well-chosen examples. At the end, an appendix about foundational functional analysis theorems is provided to help the uninitiated reader. A Guide to Spectral Theory: Applications and Exercises is intended for graduate students taking an introductory course in spectral theory or operator theory. A background in linear functional analysis and partial differential equations is assumed; basic knowledge of bounded linear operators is useful but not required. PhD students and researchers will also find this volume to be of interest, particularly the research directions provided in later chapters.
Spectral Theory
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Author : David Borthwick
language : en
Publisher: Springer Nature
Release Date : 2020-03-12
Spectral Theory written by David Borthwick and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-03-12 with Mathematics categories.
This textbook offers a concise introduction to spectral theory, designed for newcomers to functional analysis. Curating the content carefully, the author builds to a proof of the spectral theorem in the early part of the book. Subsequent chapters illustrate a variety of application areas, exploring key examples in detail. Readers looking to delve further into specialized topics will find ample references to classic and recent literature. Beginning with a brief introduction to functional analysis, the text focuses on unbounded operators and separable Hilbert spaces as the essential tools needed for the subsequent theory. A thorough discussion of the concepts of spectrum and resolvent follows, leading to a complete proof of the spectral theorem for unbounded self-adjoint operators. Applications of spectral theory to differential operators comprise the remaining four chapters. These chapters introduce the Dirichlet Laplacian operator, Schrödinger operators, operators on graphs, and the spectral theory of Riemannian manifolds. Spectral Theory offers a uniquely accessible introduction to ideas that invite further study in any number of different directions. A background in real and complex analysis is assumed; the author presents the requisite tools from functional analysis within the text. This introductory treatment would suit a functional analysis course intended as a pathway to linear PDE theory. Independent later chapters allow for flexibility in selecting applications to suit specific interests within a one-semester course.
Functional Analysis Spectral Theory
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Author : V. S. Sunder
language : en
Publisher:
Release Date : 1997
Functional Analysis Spectral Theory written by V. S. Sunder and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997 with Functional analysis categories.
Lectures given by the author at the Institute of Mathematical Sciences, Madras, in 1996.
A Short Course On Spectral Theory
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Author : William Arveson
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-04-18
A Short Course On Spectral Theory written by William Arveson 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 2006-04-18 with Mathematics categories.
This book presents the basic tools of modern analysis within the context of the fundamental problem of operator theory: to calculate spectra of specific operators on infinite dimensional spaces, especially operators on Hilbert spaces. The tools are diverse, and they provide the basis for more refined methods that allow one to approach problems that go well beyond the computation of spectra: the mathematical foundations of quantum physics, noncommutative K-theory, and the classification of simple C*-algebras being three areas of current research activity which require mastery of the material presented here.
Spectral Analysis Of Differential Operators
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Author : Fedor S. Rofe-Beketov
language : en
Publisher: World Scientific
Release Date : 2005
Spectral Analysis Of Differential Operators written by Fedor S. Rofe-Beketov and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005 with Mathematics categories.
This is the first monograph devoted to the Sturm oscillatory theory for infinite systems of differential equations and its relations with the spectral theory. It aims to study a theory of self-adjoint problems for such systems, based on an elegant method of binary relations. Another topic investigated in the book is the behavior of discrete eigenvalues which appear in spectral gaps of the Hill operator and almost periodic SchrAdinger operators due to local perturbations of the potential (e.g., modeling impurities in crystals). The book is based on results that have not been presented in other monographs. The only prerequisites needed to read it are basics of ordinary differential equations and operator theory. It should be accessible to graduate students, though its main topics are of interest to research mathematicians working in functional analysis, differential equations and mathematical physics, as well as to physicists interested in spectral theory of differential operators."
Spectral Methods In Surface Superconductivity
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Author : Søren Fournais
language : en
Publisher: Springer Science & Business Media
Release Date : 2010-06-15
Spectral Methods In Surface Superconductivity written by Søren Fournais 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 2010-06-15 with Mathematics categories.
This book examines in detail the nonlinear Ginzburg–Landau functional, the model most commonly used in the study of superconductivity. Specifically covered are cases in the presence of a strong magnetic field and with a sufficiently large Ginzburg–Landau parameter kappa. Spectral Methods in Surface Superconductivity is intended for students and researchers with a graduate-level understanding of functional analysis, spectral theory, and the analysis of partial differential equations. The book also includes an overview of all nonstandard material as well as important semi-classical techniques in spectral theory that are involved in the nonlinear study of superconductivity.
Spectral And Dynamical Stability Of Nonlinear Waves
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Author : Todd Kapitula
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-06-06
Spectral And Dynamical Stability Of Nonlinear Waves written by Todd Kapitula 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 2013-06-06 with Mathematics categories.
This book unifies the dynamical systems and functional analysis approaches to the linear and nonlinear stability of waves. It synthesizes fundamental ideas of the past 20+ years of research, carefully balancing theory and application. The book isolates and methodically develops key ideas by working through illustrative examples that are subsequently synthesized into general principles. Many of the seminal examples of stability theory, including orbital stability of the KdV solitary wave, and asymptotic stability of viscous shocks for scalar conservation laws, are treated in a textbook fashion for the first time. It presents spectral theory from a dynamical systems and functional analytic point of view, including essential and absolute spectra, and develops general nonlinear stability results for dissipative and Hamiltonian systems. The structure of the linear eigenvalue problem for Hamiltonian systems is carefully developed, including the Krein signature and related stability indices. The Evans function for the detection of point spectra is carefully developed through a series of frameworks of increasing complexity. Applications of the Evans function to the Orientation index, edge bifurcations, and large domain limits are developed through illustrative examples. The book is intended for first or second year graduate students in mathematics, or those with equivalent mathematical maturity. It is highly illustrated and there are many exercises scattered throughout the text that highlight and emphasize the key concepts. Upon completion of the book, the reader will be in an excellent position to understand and contribute to current research in nonlinear stability.
An Introduction To Spectral Theory
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Author : Andrei Giniatoulline
language : en
Publisher: R.T. Edwards, Inc.
Release Date : 2005
An Introduction To Spectral Theory written by Andrei Giniatoulline and has been published by R.T. Edwards, Inc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005 with Mathematics categories.
A brief and accessible introduction to the spectral theory of linear second order elliptic differential operators. By introducing vital topics of abstract functional analysis where necessary, and using clear and simple proofs, the book develops an elegant presentation of the theory while integrating applications of basic real world problems involving the Laplacian. Suitable for use as a self-contained introduction for beginners or as a one-semester student text; contains some 25 examples and 60 exercises, most with detailed hints.