Precise spectral asymptotics for elliptic operators acting in fiberings over manifolds with boundary
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.
scientific article; zbMATH DE number 3901231