Partial Differential Equations VII: Spectral Theory of Differential Operators (Encyclopaedia of Mathematical Sciences (64))

lgli/M_Mathematics/MRef_References/MRv_VINITI/Partial differential equations 07 (EMS64, Springer, 1994)(ISBN 3540546774)(KA)(T)(270s)_MRv_.djvu

Partial Differential Equations VII: Spectral Theory of Differential Operators (Encyclopaedia of Mathematical Sciences (64)) 🔍

G. V. Rozenblum, M. A. Shubin, M. Z. Solomyak (auth.), M. A. Shubin (eds.) Springer-Verlag Berlin Heidelberg, Encyclopaedia of Mathematical Sciences, Encyclopaedia of Mathematical Sciences 64, 7, 1, 1994

英语 [en] · DJVU · 1.9MB · 1994 · 📘 非小说类图书 · 🚀/lgli/lgrs/nexusstc/scihub/zlib · Save

描述

§18 Operators with Almost Periodic Coefficients . . . . . . . . . . . . . . . . . . . 186 18. 1. General Definitions. Essential Self-Adjointness . . . . . . . . . . . . 186 18. 2. General Properties of the Spectrum and Eigenfunctions . . . . 188 18. 3. The Spectrum of the One-Dimensional Schrödinger Operator with an Almost Periodic Potential . . . . . . . . . . . . . . 192 18. 4. The Density of States of an Operator with Almost Periodic Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 18. 5. Interpretation of the Density of States with the Aid of von Neumann Aigebras and Its Properties . . . . . . . . . . . . . . 199 §19 Operators with Random Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 206 19. 1. Translation Homogeneous Random Fields . . . . . . . . . . . . . . . . . 207 19. 2. Random DifferentialOperators . . . . . . . . . . . . . . . . . . . . . . . . . . 212 19. 3. Essential Self-Adjointness and Spectra . . . . . . . . . . . . . . . . . . . 214 19. 4. Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 19. 5. The Character of the Spectrum. Anderson Localization 220 §20 Non-Self-Adjoint Differential Operators that Are Close to Self-Adjoint Ones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 20. 1. Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 20. 2. Basic Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 20. 3. Completeness Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 20. 4. Expansion and Summability Theorems. Asymptotic Behaviour of the Spectrum . . . . . . . . . . . . . . . . . . . 228 20. 5. Application to DifferentialOperators . . . . . . . . . . . . . . . . . . . . . 230 Comments on the Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 Author Index 262 Subject Index 265 Preface The spectral theory of operators in a finite-dimensional space first appeared in connection with the description of the frequencies of small vibrations of me chanical systems (see Arnol'd et al. 1985). When the vibrations of astring are considered, there arises a simple eigenvalue problem for a differential opera tor. In the case of a homogeneous string it suffices to use the classical theory 6 Preface of Fourier series.

备用文件名

lgrsnf/M_Mathematics/MRef_References/MRv_VINITI/Partial differential equations 07 (EMS64, Springer, 1994)(ISBN 3540546774)(KA)(T)(270s)_MRv_.djvu

备用文件名

nexusstc/Partial Differential Equations VII/6ced4b34029b3e928759aa1aac537297.djvu

备用文件名

scihub/10.1007/978-3-662-06719-2.pdf

备用文件名

zlib/Computers/Computer Science/G. V. Rozenblum, M. A. Shubin, M. Z. Solomyak (auth.), M. A. Shubin (eds.)/Partial Differential Equations VII: Spectral Theory of Differential Operators_446861.djvu

备选作者

M. A. Shubin, T. Zastawniak, G. V. Rozenblum, M. Z. Solomyak

备选作者

Shubin, M. a., Zastawniak, T., Rozenblum, G. V.

备选作者

M. A Shubin; G. V Rozenblum; Margarita Solomyak

备选作者

I︠U︡. V Egorov; M. A Shubin

备选作者

M. A Shubin; Yu V Egorov

备用出版商

Spektrum Akademischer Verlag. in Springer-Verlag GmbH

备用出版商

Springer Berlin Heidelberg : Imprint : Springer

备用出版商

Springer Spektrum. in Springer-Verlag GmbH

备用出版商

Steinkopff. in Springer-Verlag GmbH

备用版本

Encyclopaedia of mathematical sciences, 64, Berlin, Heidelberg, 1994

备用版本

Encyclopaedia of Mathematical Sciences 64, Part 7, 1, 1994

备用版本

Encyclopaedia of mathematical sciences, 64, Berlin, 2011

备用版本

Springer Nature, Berlin, Heidelberg, 2013

备用版本

1 edition, December 27, 1994

备用版本

Germany, Germany

元数据中的注释

Kolxo3 -- 23

元数据中的注释

sm40107197

元数据中的注释

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备用描述

§18 Operators with Almost Periodic Coefficients . . . . . . . . . . . . . . . . . . . 186 18. 1. General Definitions. Essential Self-Adjointness . . . . . . . . . . . . 186 18. 2. General Properties of the Spectrum and Eigenfunctions . . . . 188 18. 3. The Spectrum of the One-Dimensional Schrödinger Operator with an Almost Periodic Potential . . . . . . . . . . . . . . 192 18. 4. The Density of States of an Operator with Almost Periodic Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 18. 5. Interpretation of the Density of States with the Aid of von Neumann Aigebras and Its Properties . . . . . . . . . . . . . . 199 §19 Operators with Random Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 206 19. 1. Translation Homogeneous Random Fields . . . . . . . . . . . . . . . . . 207 19. 2. Random DifferentialOperators . . . . . . . . . . . . . . . . . . . . . . . . . . 212 19. 3. Essential Self-Adjointness and Spectra . . . . . . . . . . . . . . . . . . . 214 19. 4. Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 19. 5. The Character of the Spectrum. Anderson Localization 220 §20 Non-Self-Adjoint Differential Operators that Are Close to Self-Adjoint Ones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 20. 1. Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 20. 2. Basic Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 20. 3. Completeness Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 20. 4. Expansion and Summability Theorems. Asymptotic Behaviour of the Spectrum . . . . . . . . . . . . . . . . . . . 228 20.5. Application to DifferentialOperators . . . . . . . . . . . . . . . . . . . . . 230 Comments on the Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 Author Index 262 Subject Index 265 Preface The spectral theory of operators in a finite-dimensional space first appeared in connection with the description of the frequencies of small vibrations of me chanical systems (see Arnol'd et al. 1985). When the vibrations of astring are considered, there arises a simple eigenvalue problem for a differential opera tor. In the case of a homogeneous string it suffices to use the classical theory 6 Preface of Fourier series.
Erscheinungsdatum: 06.12.2010

备用描述

§18 Operators with Almost Periodic Coefficients . . . . . . . . . . . . . . . . . . . 186 18. 1. General Definitions. Essential Self-Adjointness . . . . . . . . . . . . 186 18. 2. General Properties of the Spectrum and Eigenfunctions . . . . 188 18. 3. The Spectrum of the One-Dimensional Schrödinger Operator with an Almost Periodic Potential . . . . . . . . . . . . . . 192 18. 4. The Density of States of an Operator with Almost Periodic Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 18. 5. Interpretation of the Density of States with the Aid of von Neumann Aigebras and Its Properties . . . . . . . . . . . . . . 199 §19 Operators with Random Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 206 19. 1. Translation Homogeneous Random Fields . . . . . . . . . . . . . . . . . 207 19. 2. Random DifferentialOperators . . . . . . . . . . . . . . . . . . . . . . . . . . 212 19. 3. Essential Self-Adjointness and Spectra . . . . . . . . . . . . . . . . . . . 214 19. 4. Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 19. 5. The Character of the Spectrum. Anderson Localization 220 §20 Non-Self-Adjoint Differential Operators that Are Close to Self-Adjoint Ones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 20. 1. Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 20. 2. Basic Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 20. 3. Completeness Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 20. 4. Expansion and Summability Theorems. Asymptotic Behaviour of the Spectrum . . . . . . . . . . . . . . . . . . . 228 20.5. Application to DifferentialOperators . . . . . . . . . . . . . . . . . . . . . 230 Comments on the Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 Author Index 262 Subject Index 265 Preface The spectral theory of operators in a finite-dimensional space first appeared in connection with the description of the frequencies of small vibrations of me chanical systems (see Arnol'd et al. 1985). When the vibrations of astring are considered, there arises a simple eigenvalue problem for a differential opera tor. In the case of a homogeneous string it suffices to use the classical theory 6 Preface of Fourier series.
Erscheinungsdatum: 14.12.1994

备用描述

ʹ18 Operators with Almost Periodic Coefficients ... ... ... . 186 18. 1. General Definitions. Essential Self-Adjointness ... ... 186 18. 2. General Properties of the Spectrum and Eigenfunctions ... 188 18. 3. The Spectrum of the One-Dimensional Schrödinger Operator with an Almost Periodic Potential ... ... . . 192 18. 4. The Density of States of an Operator with Almost Periodic Coefficients ... ... ... ... ... ... ... . 197 18. 5. Interpretation of the Density of States with the Aid of von Neumann Aigebras and Its Properties ... ... . . 199 ʹ19 Operators with Random Coefficients ... ... ... ... . . 206 19. 1. Translation Homogeneous Random Fields ... ... ... . 207 19. 2. Random DifferentialOperators ... ... ... ... . . 212 19. 3. Essential Self-Adjointness and Spectra ... ... ... . 214 19. 4. Density of States ... ... ... ... ... ... . . 217 19. 5. The Character of the Spectrum. Anderson Localization 220 ʹ20 Non-Self-Adjoint Differential Operators that Are Close to Self-Adjoint Ones ... ... ... ... ... ... ... 222 20. 1. Preliminary Remarks ... ... ... ... ... ... 222 20. 2. Basic Examples ... ... ... ... ... ... ... 225 20. 3. Completeness Theorems ... ... ... ... ... . . 226 20. 4. Expansion and Summability Theorems. Asymptotic Behaviour of the Spectrum ... ... ... . 228 20. 5. Application to DifferentialOperators ... ... ... ... 230 Comments on the Literature ... ... ... ... ... ... . 234 References ... ... ... ... ... ... ... ... ... 236 Author Index 262 Subject Index 265 Preface The spectral theory of operators in a finite-dimensional space first appeared in connection with the description of the frequencies of small vibrations of meƯ chanical systems (see Arnol'd et al. 1985). When the vibrations of astring are considered, there arises a simple eigenvalue problem for a differential operaƯ tor. In the case of a homogeneous string it suffices to use the classical theory 6 Preface of Fourier series

备用描述

This EMS volume contains a survey of the principles and advanced techniques of the spectral theory of linear differential and pseudodifferential operators in finite-dimensional spaces. The basic notions and theorems are first reviewed and followed by a comprehensive presentation of a variety of advanced approaches such as the factorization method, the variational techniques, the approximate spectral projection method, and the probabilistic method, to name a few. Special attention is devoted to the spectral properties of Schrödinger and Dirac operators and of other operators as well. In addition, a special section of Sunada's recent solution of Kac's celebrated problem of whether or not "one can hear the shape of a drum" is included.

备用描述

This volume of the "Encyclopaedia of Mathematical Sciences" presents an introduction to the classical theory of partial differential equations which emphasizes physical methods and physical interpretations. Topics discussed include spectral theory, planar waves and the theory of semigroups.

备用描述

The language of the general theory of operators (mainly unbounded ones) in a Hubert space is systematically used in the spectral theory of differential operators.

备用描述

Front Matter....Pages i-v
Spectral Theory of Differential Operators....Pages 1-235
Back Matter....Pages 236-274

开源日期

2009-07-20

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