A Guide To Spectral Theory

Author : Christophe Cheverry,Nicolas Raymond
Publisher : Springer Nature
Page : 258 pages
File Size : 48,9 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 : M. Sh. Birman
Publisher : Springer Science & Business Media
Page : 96 pages
File Size : 50,8 Mb
Release : 2012-12-06
Category : Science
ISBN : 9781468475890

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Spectral Theory by M. Sh. Birman Pdf

Author : Robin Harte
Publisher : Springer
Page : 132 pages
File Size : 48,8 Mb
Release : 2014-04-29
Category : Mathematics
ISBN : 9783319056487

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Spectral Mapping Theorems by Robin Harte Pdf

Written by an author who was at the forefront of developments in multi-variable spectral theory during the seventies and the eighties, this guide sets out to describe in detail the spectral mapping theorem in one, several and many variables. The basic algebraic systems – semigroups, rings and linear algebras – are summarised, and then topological-algebraic systems, including Banach algebras, to set up the basic language of algebra and analysis. Spectral Mapping Theorems is written in an easy-to-read and engaging manner and will be useful for both the beginner and expert. It will be of great importance to researchers and postgraduates studying spectral theory.

Author : Edgar Raymond Lorch
Publisher : Unknown
Page : 184 pages
File Size : 45,6 Mb
Release : 1962
Category : Mathematics
ISBN : UOM:39015017131544

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Spectral Theory by Edgar Raymond Lorch Pdf

Author : David Eric Edmunds,W. D. Evans
Publisher : Oxford University Press
Page : 610 pages
File Size : 49,8 Mb
Release : 2018
Category : Mathematics
ISBN : 9780198812050

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Spectral Theory and Differential Operators by David Eric Edmunds,W. D. Evans Pdf

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 : Michael Sh. Birman,M.Z. Solomjak
Publisher : Springer Science & Business Media
Page : 316 pages
File Size : 47,9 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 : P. D. Hislop,I. M. Sigal
Publisher : Unknown
Page : 352 pages
File Size : 55,9 Mb
Release : 1995-11-01
Category : Electronic
ISBN : 1461207428

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

Author : Edgar R. Lorch
Publisher : Unknown
Page : 158 pages
File Size : 53,5 Mb
Release : 2003-01-01
Category : Electronic
ISBN : 0758171560

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Spectral Theory by Edgar R. Lorch Pdf

Author : John McCleary
Publisher : Cambridge University Press
Page : 579 pages
File Size : 45,7 Mb
Release : 2001
Category : Mathematics
ISBN : 9780521567596

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A User's Guide to Spectral Sequences by John McCleary Pdf

Spectral sequences are among the most elegant and powerful methods of computation in mathematics. This book describes some of the most important examples of spectral sequences and some of their most spectacular applications. The first part treats the algebraic foundations for this sort of homological algebra, starting from informal calculations. The heart of the text is an exposition of the classical examples from homotopy theory, with chapters on the Leray-Serre spectral sequence, the Eilenberg-Moore spectral sequence, the Adams spectral sequence, and, in this new edition, the Bockstein spectral sequence. The last part of the book treats applications throughout mathematics, including the theory of knots and links, algebraic geometry, differential geometry and algebra. This is an excellent reference for students and researchers in geometry, topology, and algebra.

Author : M. Sh Birman
Publisher : Unknown
Page : 99 pages
File Size : 55,5 Mb
Release : 1969
Category : Electronic
ISBN : 0608304379

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Spectral Theory by M. Sh Birman Pdf

Author : M. Sh. Birman
Publisher : Springer Science & Business Media
Page : 121 pages
File Size : 47,8 Mb
Release : 2012-12-06
Category : Science
ISBN : 9781468489262

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Spectral Theory and Wave Processes by M. Sh. Birman Pdf

Author : Kurt Otto Friedrichs
Publisher : Unknown
Page : 218 pages
File Size : 54,9 Mb
Release : 2012-07-01
Category : Electronic
ISBN : 1258444054

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Spectral Theory of Operators in Hilbert Space by Kurt Otto Friedrichs Pdf

The Present Lectures Intend To Provide An Introduction To The Spectral Analysis Of Self-Joint Operators Within The Framework Of Hilbert Space Theory. The Guiding Notion In This Approach Is That Of Spectral Representation. At The Same Time The Notion Of Function Of An Operator Is Emphasized. The Definition Of Hilbert Space: In Mathematics, A Hilbert Space Is A Real Or Complex Vector Space With A Positive-Definite Hermitian Form, That Is Complete Under Its Norm. Thus It Is An Inner Product Space, Which Means That It Has Notions Of Distance And Of Angle (Especially The Notion Of Orthogonality Or Perpendicularity). The Completeness Requirement Ensures That For Infinite Dimensional Hilbert Spaces The Limits Exist When Expected, Which Facilitates Various Definitions From Calculus. A Typical Example Of A Hilbert Space Is The Space Of Square Summable Sequences. Hilbert Spaces Allow Simple Geometric Concepts, Like Projection And Change Of Basis To Be Applied To Infinite Dimensional Spaces, Such As Function Spaces. They Provide A Context With Which To Formalize And Generalize The Concepts Of The Fourier Series In Terms Of Arbitrary Orthogonal Polynomials And Of The Fourier Transform, Which Are Central Concepts From Functional Analysis. Hilbert Spaces Are Of Crucial Importance In The Mathematical Formulation Of Quantum Mechanics.

Author : Hiroshi Isozaki
Publisher : Springer Nature
Page : 130 pages
File Size : 52,6 Mb
Release : 2020-09-26
Category : Science
ISBN : 9789811581991

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Inverse Spectral and Scattering Theory by Hiroshi Isozaki Pdf

The aim of this book is to provide basic knowledge of the inverse problems arising in various areas in mathematics, physics, engineering, and medical science. These practical problems boil down to the mathematical question in which one tries to recover the operator (coefficients) or the domain (manifolds) from spectral data. The characteristic properties of the operators in question are often reduced to those of Schrödinger operators. We start from the 1-dimensional theory to observe the main features of inverse spectral problems and then proceed to multi-dimensions. The first milestone is the Borg–Levinson theorem in the inverse Dirichlet problem in a bounded domain elucidating basic motivation of the inverse problem as well as the difference between 1-dimension and multi-dimension. The main theme is the inverse scattering, in which the spectral data is Heisenberg’s S-matrix defined through the observation of the asymptotic behavior at infinity of solutions. Significant progress has been made in the past 30 years by using the Faddeev–Green function or the complex geometrical optics solution by Sylvester and Uhlmann, which made it possible to reconstruct the potential from the S-matrix of one fixed energy. One can also prove the equivalence of the knowledge of S-matrix and that of the Dirichlet-to-Neumann map for boundary value problems in bounded domains. We apply this idea also to the Dirac equation, the Maxwell equation, and discrete Schrödinger operators on perturbed lattices. Our final topic is the boundary control method introduced by Belishev and Kurylev, which is for the moment the only systematic method for the reconstruction of the Riemannian metric from the boundary observation, which we apply to the inverse scattering on non-compact manifolds. We stress that this book focuses on the lucid exposition of these problems and mathematical backgrounds by explaining the basic knowledge of functional analysis and spectral theory, omitting the technical details in order to make the book accessible to graduate students as an introduction to partial differential equations (PDEs) and functional analysis.

Author : Bernard Helffer
Publisher : Cambridge University Press
Page : 263 pages
File Size : 50,6 Mb
Release : 2013-01-17
Category : Mathematics
ISBN : 9781107032309

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Spectral Theory and Its Applications by Bernard Helffer Pdf

Introduces the basic tools in spectral analysis using numerous examples from the Schrödinger operator theory and various branches of physics.

Author : R. Carmona,J. Lacroix
Publisher : Springer Science & Business Media
Page : 611 pages
File Size : 55,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461244882

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Spectral Theory of Random Schrödinger Operators by R. Carmona,J. Lacroix Pdf

Since the seminal work of P. Anderson in 1958, localization in disordered systems has been the object of intense investigations. Mathematically speaking, the phenomenon can be described as follows: the self-adjoint operators which are used as Hamiltonians for these systems have a ten dency to have pure point spectrum, especially in low dimension or for large disorder. A lot of effort has been devoted to the mathematical study of the random self-adjoint operators relevant to the theory of localization for disordered systems. It is fair to say that progress has been made and that the un derstanding of the phenomenon has improved. This does not mean that the subject is closed. Indeed, the number of important problems actually solved is not larger than the number of those remaining. Let us mention some of the latter: • A proof of localization at all energies is still missing for two dimen sional systems, though it should be within reachable range. In the case of the two dimensional lattice, this problem has been approached by the investigation of a finite discrete band, but the limiting pro cedure necessary to reach the full two-dimensional lattice has never been controlled. • The smoothness properties of the density of states seem to escape all attempts in dimension larger than one. This problem is particularly serious in the continuous case where one does not even know if it is continuous.