Ebook: On Spectral Theory of Elliptic Operators

It is well known that a wealth of problems of different nature, applied as well as purely theoretic, can be reduced to the study of elliptic equations and their eigen-values. During the years many books and articles have been published on this topic, considering spectral properties of elliptic differential operators from different points of view. This is one more book on these properties. This book is devoted to the study of some classical problems of the spectral theory of elliptic differential equations. The reader will find hardly any intersections with the books of Shubin [Sh] or Rempel-Schulze [ReSch] or with the works cited there. This book also has no general information in common with the books by Egorov and Shubin [EgShu], which also deal with spectral properties of elliptic operators. There is nothing here on oblique derivative problems; the reader will meet no pseudodifferential operators. The main subject of the book is the estimates of eigenvalues, especially of the first one, and of eigenfunctions of elliptic operators. The considered problems have in common the approach consisting of the application of the variational principle and some a priori estimates, usually in Sobolev spaces. In many cases, impor­ tant for physics and mechanics, as well as for geometry and analysis, this rather elementary approach allows one to obtain sharp results.

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This book contains results obtained by the authors in 1980-1992 and partially published in mathematical journals; some of them have not been translated into English before. The results relate to estimates of the first eigenvalue and the negative spectrum of the Schr?dinger operator, as well as to estimates of eigenvalues and eigenfunctions. The first part of the book is of introductory character and contains the required tools from the theory of functional spaces, spectral theory, and the theory of elliptic boundary-value problems. It is essentially self-contained and can be used as a textbook by students.

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This book contains results obtained by the authors in 1980-1992 and partially published in mathematical journals; some of them have not been translated into English before. The results relate to estimates of the first eigenvalue and the negative spectrum of the Schr?dinger operator, as well as to estimates of eigenvalues and eigenfunctions. The first part of the book is of introductory character and contains the required tools from the theory of functional spaces, spectral theory, and the theory of elliptic boundary-value problems. It is essentially self-contained and can be used as a textbook by students.
Content:
Front Matter....Pages I-X
Hilbert Spaces....Pages 1-24
Functional Spaces....Pages 25-107
Elliptic Operators....Pages 109-131
Spectral Properties of Elliptic Operators....Pages 133-151
The Sturm-Liouville Problem....Pages 153-206
Differential Operators of Any Order....Pages 207-232
Eigenfunctions of Elliptic Operators in Bounded Domains....Pages 233-273
Negative Spectra of Elliptic Operators....Pages 275-317
Back Matter....Pages 319-332

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This book contains results obtained by the authors in 1980-1992 and partially published in mathematical journals; some of them have not been translated into English before. The results relate to estimates of the first eigenvalue and the negative spectrum of the Schr?dinger operator, as well as to estimates of eigenvalues and eigenfunctions. The first part of the book is of introductory character and contains the required tools from the theory of functional spaces, spectral theory, and the theory of elliptic boundary-value problems. It is essentially self-contained and can be used as a textbook by students.
Content:
Front Matter....Pages I-X
Hilbert Spaces....Pages 1-24
Functional Spaces....Pages 25-107
Elliptic Operators....Pages 109-131
Spectral Properties of Elliptic Operators....Pages 133-151
The Sturm-Liouville Problem....Pages 153-206
Differential Operators of Any Order....Pages 207-232
Eigenfunctions of Elliptic Operators in Bounded Domains....Pages 233-273
Negative Spectra of Elliptic Operators....Pages 275-317
Back Matter....Pages 319-332
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