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

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

          
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About the Book

§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.

Table of Contents:
Contents: Spectral Theory of Differential Operators by G.V. Rozenblum, M.A. Shubin, M.Z. Solomyak.


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Product Details
  • ISBN-13: 9783540546771
  • Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
  • Publisher Imprint: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Edition: 1994 ed.
  • Language: English
  • Returnable: Y
  • Spine Width: 18 mm
  • Weight: 575 gr
  • ISBN-10: 3540546774
  • Publisher Date: 14 Dec 1994
  • Binding: Hardback
  • Height: 235 mm
  • No of Pages: 274
  • Series Title: 64 Encyclopaedia of Mathematical Sciences
  • Sub Title: Spectral Theory of Differential Operators
  • Width: 155 mm


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