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The Asymptotic Distribution of Eigenvalues of Partial Differential Operators
Yu. Safarov, King's College, London, England, and D. Vassiliev, University of Sussex, Falmer Brighton, England
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Translations of Mathematical Monographs
1997; 354 pp; softcover
Volume: 155
Reprint/Revision History:
reprinted with corrections 1998
ISBN-10: 0-8218-0921-0
ISBN-13: 978-0-8218-0921-1
List Price: US$122
Member Price: US$97.60
Order Code: MMONO/155.S
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As the subject of extensive research for over a century, spectral asymptotics for partial differential operators attracted the attention of many outstanding mathematicians and physicists. This book studies the eigenvalues of elliptic linear boundary value problems and has as its main content a collection of asymptotic formulas describing the distribution of eigenvalues with high sequential numbers. Asymptotic formulas are used to illustrate standard eigenvalue problems of mechanics and mathematical physics.

The volume provides a basic introduction to all the necessary mathematical concepts and tools, such as microlocal analysis, billiards, symplectic geometry and Tauberian theorems. It is self-contained and would be suitable as a graduate text.

Readership

Graduate students, research mathematicians, applied mathematicians, engineers, and physicists interested in partial differential equations.

Reviews

"In the reviewer's opinion, this book is indispensable for serious students of spectral asymptotics."

-- Lars Hörmander for the Bulletin of the London Mathematical Society

Table of Contents

  • Main results
  • Oscillatory integrals
  • Construction of the wave group
  • Singularities of the wave group
  • Proof of main results
  • Mechanical applications
  • Appendix A. Spectral problem on the half-line
  • Appendix B. Fourier Tauberian theorems
  • Appendix C. Stationary phase formula
  • Appendix D. Hamiltonian billiards: proofs
  • Appendix E. Factorization of smooth functions and Taylor-type formulae
  • References
  • Principal notation
  • Index
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