Precise spectral asymptotics for elliptic operators acting in fiberings over manifolds with boundary
type: Item
instance of: scholarly article
title: Precise spectral asymptotics for elliptic operators acting in fiberings over manifolds with boundary
zbMATH Open document ID: 0565.35002
published in: Lecture Notes in Mathematics
publication date: 1984-01-01T00:00:00Z
review text: The book studies the asymptotics for large k of the eigenvalue distribution function N(k), i.e. of the number of eigenvalues between O and \(k^ 2\), for elliptic selfadjoint operators on fiber bundles over compact manifolds with boundary. It summarizes the series of papers which the author published during the last five years. It has already become common to derive spectral properties of elliptic operators from spectral properties of corresponding hyperbolic operators. Then the first term of the asymptotical expansion of N(k) can be obtained by application of the Tauberian theorems to certain oscillatory integrals. To obtain the second term of the asymptotics one has to study the propagation of singularities of solutions of corresponding hyperbolic equations. When the initial elliptic operator is the Laplace-Beltrami operator then geodesic flows with reflection at the boundary comes into the picture. Similar results about the asymptotics of N(k) were obtained recently by R. Melrose. This book is the first one devoted entirely to spectral asymptotics for elliptic operators. The reader can find a unified exposition of results which were previously scattered in different journals. Although this book is not an easy reading it will be a valuable source for mathematicians working in the area of partial differential equations.
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 58J45
zbMATH DE Number: 3901231
zbMATH Keywords: asymptotics
zbMATH Keywords: eigenvalue distribution function
zbMATH Keywords: selfadjoint operators
zbMATH Keywords: fiber bundles
zbMATH Keywords: compact manifolds
zbMATH Keywords: Tauberian theorems
zbMATH Keywords: oscillatory integrals
zbMATH Keywords: propagation of singularities
zbMATH Keywords: Laplace-Beltrami operator
author: Victor Ivrii
reviewed by: Yuri Kifer
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0072205
OpenAlex ID: W600786134
DOI: 10.1007/BFB0072205
description: scientific article; zbMATH DE number 3901231