Microlocal Analysis and Precise Spectral Asymptotics

Ivrii, Victor (University of Toronto, Canada)

ISBN 10: 3540627804 / ISBN 13: 9783540627807

Published by Springer-Verlag Berlin and Heidelberg GmbH & Co. KG, 1998

This text provides methods of microlocal semiclassical analysis in applications to spectral asymptotics with accurate remainder estimates. The machinery of local and microlocal semiclassical spectral asymptotics is developed, as are methods in combining these asymptotics with spectral estimates. Series: Springer Monographs in Mathematics. Num Pages: 748 pages, biography. BIC Classification: PBKF. Category: (P) Professional & Vocational; (UP) Postgraduate, Research & Scholarly. Dimension: 234 x 156 x 39. Weight in Grams: 1226. . 1998. Hardback. . . . . Seller Inventory # V9783540627807

Synopsis:

The problem of spectral asymptotics, in particular the problem of the asymptotic dis­ tribution of eigenvalues, is one of the central problems in the spectral theory of partial differential operators; moreover, it is very important for the general theory of partial differential operators. I started working in this domain in 1979 after R. Seeley found a remainder estimate of the same order as the then hypothetical second term for the Laplacian in domains with boundary, and M. Shubin and B. M. Levitan suggested that I should try to prove Weyl's conjecture. During the past fifteen years I have not left the topic, although I had such intentions in 1985 when the methods I invented seemed to fai! to provide furt her progress and only a couple of not very exciting problems remained to be solved. However, at that time I made the step toward local semiclassical spectral asymptotics and rescaling, and new horizons opened.

From the Back Cover: Devoted to the methods of microlocal analysis applied to spectral asymptotics with accurate remainder estimates, this long awaited book develops the very powerful machinery of local and microlocal semiclassical spectral asymptotics, as well as methods of combining these asymptotics with spectral estimates. The rescaling technique, an easy to use and very powerful tool, is presented. Many theorems, considered till now as independent and difficult, are now just special cases of easy corollaries of the theorems proved in this book. Most of the results and their proofs are as yet unpublished. Part 1 considers semiclassical microlocal analysis and propagation of singularities inside the domain and near the boundary. Part 2 is on local and microlocal semiclassical spectral asymptotics for general operators and Schrödinger and Dirac operators. After a synthesis in Part 3, the real fun begins in Part 4: the main theorems are applied and numerous results, both known and new, are recovered with little effort. Then, in Chapter 12, non-Weyl asymptotics are obtained for operators in domains with thick cusps, degenerate operators, for spectral Riesz means for operators with singularities. Most of the results and almost all the proofs were never published.

Bibliographic Details

Title: Microlocal Analysis and Precise Spectral ...
Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Publication Date: 1998
Binding: Hardcover
Condition: New