Propagation of Singularities near the Boundary
In this chapter we perform the same task as in the previous chapter but near the smooth boundary. In section 3.1 we prove the basic theorem which is formulated in terms of auxiliary functions as was theorem 2.1.2. In section 3.2 we prove some statements s
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Springer-Verlag Berlin Heidelberg GmbH
Victor Ivrii
Microlocal Analysis and Precise Spectral Asym ptotics
,
Springer
Victor lvrii Department of Mathematics University of Toronto WO, St. George Street Toronto, Ontario M5S lAI
Canada
Llbrary of Congress Cataloglng-ln-Publlcatlon Data
Ivr 11, Vlctor, 1949Mlcrolocal analysIs and preelse speetral asy.ptotles I Vletor Ivr 11, p, em. -- (SprInger .onographs In mathe.atles) Ineludes blbllographleal referenees and Index. ISBN 978-3-642-08307-5 ISBN 978-3-662-12496-3 (eBook) DOI 10.1007/978-3-662-12496-3 1. Mleroloeal analysIs. 2. Spectral theory (Mathe.atles) 3. Elgenvalues. 4. Asymptotle expansIons. 1. Tlt1e. II. Serles. OA299.6.187 1998 515'.7242--de21 97-49029 CIP
Mathematics Subject Classification (I 991): 35P, 35J, 35L, 58G, 81Q
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Preface
The problem of spectral asymptotics, in particular the problem of the asymptotic distribution 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. So I can say that this book is the result of fifteen years of work in the theory of spectral asymptotics. However, it would be more accurate to introduce it as the result of eight years of great effort, since during the
