Operator Calculus And Spectral Theory

Author : Christophe Cheverry,Nicolas Raymond
Publisher : Springer Nature
Page : 258 pages
File Size : 42,6 Mb
Release : 2021-05-06
Category : Mathematics
ISBN : 9783030674625

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A Guide to Spectral Theory by Christophe Cheverry,Nicolas Raymond Pdf

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.

Author : Henry R. Dowson
Publisher : Unknown
Page : 444 pages
File Size : 52,5 Mb
Release : 1978
Category : Banach spaces
ISBN : UCAL:B4406582

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Spectral Theory of Linear Operators by Henry R. Dowson Pdf

General spectral theory; Riesz operators; Hermitian operators; Prespectral operators; Well-bounded operators.

Author : David Edmunds,Des Evans
Publisher : Oxford University Press
Page : 128 pages
File Size : 49,7 Mb
Release : 2018-05-03
Category : Mathematics
ISBN : 9780192540102

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Spectral Theory and Differential Operators by David Edmunds,Des Evans Pdf

This book is an updated version of the classic 1987 monograph "Spectral Theory and Differential Operators".The original book was a cutting edge account of the theory of bounded and closed linear operators in Banach and Hilbert spaces relevant to spectral problems involving differential equations. It is accessible to a graduate student as well as meeting the needs of seasoned researchers in mathematics and mathematical physics. This revised edition corrects various errors, and adds extensive notes to the end of each chapter which describe the considerable progress that has been made on the topic in the last 30 years.

Author : M. Demuth,B.W. Schulze,B. Gramsch
Publisher : Birkhäuser
Page : 355 pages
File Size : 55,5 Mb
Release : 2012-12-06
Category : Science
ISBN : 9783034886239

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Operator Calculus and Spectral Theory by M. Demuth,B.W. Schulze,B. Gramsch Pdf

Author : Carlos S. Kubrusly
Publisher : Springer Nature
Page : 249 pages
File Size : 46,9 Mb
Release : 2020-01-30
Category : Mathematics
ISBN : 9783030331498

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Spectral Theory of Bounded Linear Operators by Carlos S. Kubrusly Pdf

This textbook introduces spectral theory for bounded linear operators by focusing on (i) the spectral theory and functional calculus for normal operators acting on Hilbert spaces; (ii) the Riesz-Dunford functional calculus for Banach-space operators; and (iii) the Fredholm theory in both Banach and Hilbert spaces. Detailed proofs of all theorems are included and presented with precision and clarity, especially for the spectral theorems, allowing students to thoroughly familiarize themselves with all the important concepts. Covering both basic and more advanced material, the five chapters and two appendices of this volume provide a modern treatment on spectral theory. Topics range from spectral results on the Banach algebra of bounded linear operators acting on Banach spaces to functional calculus for Hilbert and Banach-space operators, including Fredholm and multiplicity theories. Supplementary propositions and further notes are included as well, ensuring a wide range of topics in spectral theory are covered. Spectral Theory of Bounded Linear Operators is ideal for graduate students in mathematics, and will also appeal to a wider audience of statisticians, engineers, and physicists. Though it is mostly self-contained, a familiarity with functional analysis, especially operator theory, will be helpful.

Author : Christiane Tretter
Publisher : World Scientific
Page : 297 pages
File Size : 53,9 Mb
Release : 2008
Category : Science
ISBN : 9781860947681

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Spectral Theory of Block Operator Matrices and Applications by Christiane Tretter Pdf

This book presents new concepts in operator theory and covers classes of operators (in particular, non-selfadjoint operators) which exhibit various interesting phenomena. Special attention is paid to applications in many areas of mathematical physics, including quantum mechanics, fluid mechanics, and magnetohydrodynamics.The author also discusses an operator theoretic approach to spectral problems for linear operators admitting a certain block structure. The results apply to bounded or finite-dimensional operators like block matrices as well to unbounded operators describing systems of differential equations. New concepts of numerical range are developed.

Author : Michael Sh. Birman,M.Z. Solomjak
Publisher : Springer Science & Business Media
Page : 316 pages
File Size : 48,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789400945869

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Spectral Theory of Self-Adjoint Operators in Hilbert Space by Michael Sh. Birman,M.Z. Solomjak Pdf

It isn't that they can't see the solution. It is Approach your problems from the right end that they can't see the problem. and begin with the answers. Then one day, perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be com pletely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order" , which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

Author : Joachim Weidmann
Publisher : Springer
Page : 310 pages
File Size : 54,7 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540479123

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Spectral Theory of Ordinary Differential Operators by Joachim Weidmann Pdf

These notes will be useful and of interest to mathematicians and physicists active in research as well as for students with some knowledge of the abstract theory of operators in Hilbert spaces. They give a complete spectral theory for ordinary differential expressions of arbitrary order n operating on -valued functions existence and construction of self-adjoint realizations via boundary conditions, determination and study of general properties of the resolvent, spectral representation and spectral resolution. Special attention is paid to the question of separated boundary conditions, spectral multiplicity and absolutely continuous spectrum. For the case nm=2 (Sturm-Liouville operators and Dirac systems) the classical theory of Weyl-Titchmarch is included. Oscillation theory for Sturm-Liouville operators and Dirac systems is developed and applied to the study of the essential and absolutely continuous spectrum. The results are illustrated by the explicit solution of a number of particular problems including the spectral theory one partical Schrödinger and Dirac operators with spherically symmetric potentials. The methods of proof are functionally analytic wherever possible.

Author : E. Brian Davies
Publisher : Cambridge University Press
Page : 198 pages
File Size : 55,5 Mb
Release : 1995
Category : Mathematics
ISBN : 0521587107

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Spectral Theory and Differential Operators by E. Brian Davies Pdf

This book could be used either for self-study or as a course text, and aims to lead the reader to the more advanced literature on partial differential operators.

Author : Fabrizio Colombo,Jonathan Gantner,David P. Kimsey
Publisher : Springer
Page : 356 pages
File Size : 42,9 Mb
Release : 2019-01-04
Category : Mathematics
ISBN : 9783030030742

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Spectral Theory on the S-Spectrum for Quaternionic Operators by Fabrizio Colombo,Jonathan Gantner,David P. Kimsey Pdf

The subject of this monograph is the quaternionic spectral theory based on the notion of S-spectrum. With the purpose of giving a systematic and self-contained treatment of this theory that has been developed in the last decade, the book features topics like the S-functional calculus, the F-functional calculus, the quaternionic spectral theorem, spectral integration and spectral operators in the quaternionic setting. These topics are based on the notion of S-spectrum of a quaternionic linear operator. Further developments of this theory lead to applications in fractional diffusion and evolution problems that will be covered in a separate monograph.

Author : P.D. Hislop,I.M. Sigal
Publisher : Springer Science & Business Media
Page : 331 pages
File Size : 53,9 Mb
Release : 2012-12-06
Category : Technology & Engineering
ISBN : 9781461207412

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Introduction to Spectral Theory by P.D. Hislop,I.M. Sigal Pdf

The intention of this book is to introduce students to active areas of research in mathematical physics in a rather direct way minimizing the use of abstract mathematics. The main features are geometric methods in spectral analysis, exponential decay of eigenfunctions, semi-classical analysis of bound state problems, and semi-classical analysis of resonance. A new geometric point of view along with new techniques are brought out in this book which have both been discovered within the past decade. This book is designed to be used as a textbook, unlike the competitors which are either too fundamental in their approach or are too abstract in nature to be considered as texts. The authors' text fills a gap in the marketplace.

Author : L. Boutet de Monvel,Victor Guillemin
Publisher : Princeton University Press
Page : 166 pages
File Size : 51,8 Mb
Release : 2016-03-02
Category : Mathematics
ISBN : 9781400881444

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The Spectral Theory of Toeplitz Operators. (AM-99), Volume 99 by L. Boutet de Monvel,Victor Guillemin Pdf

The theory of Toeplitz operators has come to resemble more and more in recent years the classical theory of pseudodifferential operators. For instance, Toeplitz operators possess a symbolic calculus analogous to the usual symbolic calculus, and by symbolic means one can construct parametrices for Toeplitz operators and create new Toeplitz operators out of old ones by functional operations. If P is a self-adjoint pseudodifferential operator on a compact manifold with an elliptic symbol that is of order greater than zero, then it has a discrete spectrum. Also, it is well known that the asymptotic behavior of its eigenvalues is closely related to the behavior of the bicharacteristic flow generated by its symbol. It is natural to ask if similar results are true for Toeplitz operators. In the course of answering this question, the authors explore in depth the analogies between Toeplitz operators and pseudodifferential operators and show that both can be viewed as the "quantized" objects associated with functions on compact contact manifolds.

Author : César R. de Oliveira
Publisher : Springer Science & Business Media
Page : 410 pages
File Size : 42,7 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 : Carlos S. Kubrusly
Publisher : Springer Science & Business Media
Page : 203 pages
File Size : 44,9 Mb
Release : 2012-06-01
Category : Mathematics
ISBN : 9780817683283

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Spectral Theory of Operators on Hilbert Spaces by Carlos S. Kubrusly Pdf

This work is a concise introduction to spectral theory of Hilbert space operators. Its emphasis is on recent aspects of theory and detailed proofs, with the primary goal of offering a modern introductory textbook for a first graduate course in the subject. The coverage of topics is thorough, as the book explores various delicate points and hidden features often left untreated. Spectral Theory of Operators on Hilbert Spaces is addressed to an interdisciplinary audience of graduate students in mathematics, statistics, economics, engineering, and physics. It will also be useful to working mathematicians using spectral theory of Hilbert space operators, as well as for scientists wishing to apply spectral theory to their field. ​

Author : S Kantorovitz
Publisher : CRC Press
Page : 156 pages
File Size : 43,6 Mb
Release : 1995-06-19
Category : Mathematics
ISBN : 0582277787

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Semigroups of Operators and Spectral Theory by S Kantorovitz Pdf

This book presents some aspects of the theory of semigroups of operators, mostly from the point of view of its interaction withspectral theory. In order to make it self-contained, a concise description of the basic theory of semigroups, with complete proofs, is included in Part I. Some of the author's recent results, such as the construction of the Hille-Yosida space for general operators, the semi-simplicity manifold, and a Taylor formula for semigroups as functions of their generator, are also included in Part I. Part II describes recent generalizations (most of them in bookform for the first time), including pre-semigroups, semi-simplicity manifolds in situations more general than that considered in Part I, semigroups of unbounded symmetric operators, and an analogous result on "local cosine families" and semi-analytic vectors. It is hoped that this book will inspire more research in this field. This book will be of particular interest to graduate students and researchers working operator theory and its applications.