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Pseudodifferential Operators and Spectral Theory

Author: M.A. Shubin

Publisher: Springer Science & Business Media

Published: 2011-06-28

Total Pages: 296

ISBN-13: 3642565794

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I had mixed feelings when I thought how I should prepare the book for the second edition. It was clear to me that I had to correct all mistakes and misprints that were found in the book during the life of the first edition. This was easy to do because the mistakes were mostly minor and easy to correct, and the misprints were not many. It was more difficult to decide whether I should update the book (or at least its bibliography) somehow. I decided that it did not need much of an updating. The main value of any good mathematical book is that it teaches its reader some language and some skills. It can not exhaust any substantial topic no matter how hard the author tried. Pseudodifferential operators became a language and a tool of analysis of partial differential equations long ago. Therefore it is meaningless to try to exhaust this topic. Here is an easy proof. As of July 3, 2000, MathSciNet (the database of the American Mathematical Society) in a few seconds found 3695 sources, among them 363 books, during its search for "pseudodifferential operator". (The search also led to finding 963 sources for "pseudo-differential operator" but I was unable to check how much the results ofthese two searches intersected). This means that the corresponding words appear either in the title or in the review published in Mathematical Reviews.

Book Synopsis Pseudodifferential Operators and Spectral Theory by : M.A. Shubin

Download or read book Pseudodifferential Operators and Spectral Theory written by M.A. Shubin and published by Springer Science & Business Media. This book was released on 2011-06-28 with total page 296 pages. Available in PDF, EPUB and Kindle. Book excerpt: I had mixed feelings when I thought how I should prepare the book for the second edition. It was clear to me that I had to correct all mistakes and misprints that were found in the book during the life of the first edition. This was easy to do because the mistakes were mostly minor and easy to correct, and the misprints were not many. It was more difficult to decide whether I should update the book (or at least its bibliography) somehow. I decided that it did not need much of an updating. The main value of any good mathematical book is that it teaches its reader some language and some skills. It can not exhaust any substantial topic no matter how hard the author tried. Pseudodifferential operators became a language and a tool of analysis of partial differential equations long ago. Therefore it is meaningless to try to exhaust this topic. Here is an easy proof. As of July 3, 2000, MathSciNet (the database of the American Mathematical Society) in a few seconds found 3695 sources, among them 363 books, during its search for "pseudodifferential operator". (The search also led to finding 963 sources for "pseudo-differential operator" but I was unable to check how much the results ofthese two searches intersected). This means that the corresponding words appear either in the title or in the review published in Mathematical Reviews.

Pseudodifferential Operators and Spectral Theory

Author: Mikhail A. Shubin

Publisher: Springer

Published: 1987

Total Pages: 0

ISBN-13: 9783642968549

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The theory of pseudo differential operators (abbreviated PD~) is compara tively young; in its modern form it was created in the mid-sixties. The progress achieved with its help, however, has been so essential that without PD~ it would indeed be difficult to picture modern analysis and mathematical physics. PD~ are of particular importance in the study of elliptic equations. Even the simplest operations on elliptic operators (e. g. taking the inverse or the square root) lead out of the class of differential operators but will, under reasonable assumptions, preserve the class of PD~. A significant role is played by PD~ in the index theory for elliptic operators, where PD~ are needed to extend the class of possible deformations of an operator. PD~ appear naturally in the reduction to the boundary for any elliptic boundary problem. In this way, PD~ arise not as an end-in-themselves, but as a powerful and natural tool for the study of partial differential operators (first and foremost elliptic and hypo elliptic ones). In many cases, PD~ allow us not only to establish new theorems but also to have a fresh look at old ones and thereby obtain simpler and more transparent formulations of already known facts. This is, for instance, the case in the theory of Sobolev spaces. A natural generalization of PD~ are the Fourier integral operators (abbreviatedFIO), the first version ofwhich was the Maslov canonical operator.

Book Synopsis Pseudodifferential Operators and Spectral Theory by : Mikhail A. Shubin

Download or read book Pseudodifferential Operators and Spectral Theory written by Mikhail A. Shubin and published by Springer. This book was released on 1987 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of pseudo differential operators (abbreviated PD~) is compara tively young; in its modern form it was created in the mid-sixties. The progress achieved with its help, however, has been so essential that without PD~ it would indeed be difficult to picture modern analysis and mathematical physics. PD~ are of particular importance in the study of elliptic equations. Even the simplest operations on elliptic operators (e. g. taking the inverse or the square root) lead out of the class of differential operators but will, under reasonable assumptions, preserve the class of PD~. A significant role is played by PD~ in the index theory for elliptic operators, where PD~ are needed to extend the class of possible deformations of an operator. PD~ appear naturally in the reduction to the boundary for any elliptic boundary problem. In this way, PD~ arise not as an end-in-themselves, but as a powerful and natural tool for the study of partial differential operators (first and foremost elliptic and hypo elliptic ones). In many cases, PD~ allow us not only to establish new theorems but also to have a fresh look at old ones and thereby obtain simpler and more transparent formulations of already known facts. This is, for instance, the case in the theory of Sobolev spaces. A natural generalization of PD~ are the Fourier integral operators (abbreviatedFIO), the first version ofwhich was the Maslov canonical operator.

Pseudodifferential Operators and Spectral Theory

Author: Mikhail Aleksandrovich Shubin

Publisher: Springer

Published: 1987

Total Pages: 278

ISBN-13: 9783540136217

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Book Synopsis Pseudodifferential Operators and Spectral Theory by : Mikhail Aleksandrovich Shubin

Download or read book Pseudodifferential Operators and Spectral Theory written by Mikhail Aleksandrovich Shubin and published by Springer. This book was released on 1987 with total page 278 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Partial Differential Equations VII

Author: M.A. Shubin

Publisher: Springer Science & Business Media

Published: 2013-03-09

Total Pages: 278

ISBN-13: 3662067196

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This EMS volume contains a survey of the principles and advanced techniques of the spectral theory of linear differential and pseudodifferential operators in finite-dimensional spaces. Also including a special section of Sunada's recent solution of Kac's celebrated problem of whether or not "one can hear the shape of a drum".

Book Synopsis Partial Differential Equations VII by : M.A. Shubin

Download or read book Partial Differential Equations VII written by M.A. Shubin and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 278 pages. Available in PDF, EPUB and Kindle. Book excerpt: This EMS volume contains a survey of the principles and advanced techniques of the spectral theory of linear differential and pseudodifferential operators in finite-dimensional spaces. Also including a special section of Sunada's recent solution of Kac's celebrated problem of whether or not "one can hear the shape of a drum".

Pseudodifferential Operators (PMS-34)

Author: Michael Eugene Taylor

Publisher: Princeton University Press

Published: 2017-03-14

Total Pages: 464

ISBN-13: 1400886104

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Here Michael Taylor develops pseudodifferential operators as a tool for treating problems in linear partial differential equations, including existence, uniqueness, and estimates of smoothness, as well as other qualitative properties. Originally published in 1981. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Book Synopsis Pseudodifferential Operators (PMS-34) by : Michael Eugene Taylor

Download or read book Pseudodifferential Operators (PMS-34) written by Michael Eugene Taylor and published by Princeton University Press. This book was released on 2017-03-14 with total page 464 pages. Available in PDF, EPUB and Kindle. Book excerpt: Here Michael Taylor develops pseudodifferential operators as a tool for treating problems in linear partial differential equations, including existence, uniqueness, and estimates of smoothness, as well as other qualitative properties. Originally published in 1981. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

The Spectral Theory of Toeplitz Operators

Author: L. Boutet de Monvel

Publisher: Princeton University Press

Published: 1981-08-21

Total Pages: 172

ISBN-13: 9780691082790

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The theory of Toeplitz operators has come to resemble more and more in recent years the classical theory of pseudodifferential operators. For instance, Toeplitz operators possess a symbolic calculus analogous to the usual symbolic calculus, and by symbolic means one can construct parametrices for Toeplitz operators and create new Toeplitz operators out of old ones by functional operations. If P is a self-adjoint pseudodifferential operator on a compact manifold with an elliptic symbol that is of order greater than zero, then it has a discrete spectrum. Also, it is well known that the asymptotic behavior of its eigenvalues is closely related to the behavior of the bicharacteristic flow generated by its symbol. It is natural to ask if similar results are true for Toeplitz operators. In the course of answering this question, the authors explore in depth the analogies between Toeplitz operators and pseudodifferential operators and show that both can be viewed as the "quantized" objects associated with functions on compact contact manifolds.

Book Synopsis The Spectral Theory of Toeplitz Operators by : L. Boutet de Monvel

Download or read book The Spectral Theory of Toeplitz Operators written by L. Boutet de Monvel and published by Princeton University Press. This book was released on 1981-08-21 with total page 172 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of Toeplitz operators has come to resemble more and more in recent years the classical theory of pseudodifferential operators. For instance, Toeplitz operators possess a symbolic calculus analogous to the usual symbolic calculus, and by symbolic means one can construct parametrices for Toeplitz operators and create new Toeplitz operators out of old ones by functional operations. If P is a self-adjoint pseudodifferential operator on a compact manifold with an elliptic symbol that is of order greater than zero, then it has a discrete spectrum. Also, it is well known that the asymptotic behavior of its eigenvalues is closely related to the behavior of the bicharacteristic flow generated by its symbol. It is natural to ask if similar results are true for Toeplitz operators. In the course of answering this question, the authors explore in depth the analogies between Toeplitz operators and pseudodifferential operators and show that both can be viewed as the "quantized" objects associated with functions on compact contact manifolds.

Partial Differential Equations

Author:

Publisher:

Published: 1991

Total Pages:

ISBN-13: 9780387546773

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Book Synopsis Partial Differential Equations by :

Download or read book Partial Differential Equations written by and published by . This book was released on 1991 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Partial Differential Equations VII

Author: M.A. Shubin

Publisher: Springer

Published: 2012-12-22

Total Pages: 274

ISBN-13: 9783662067208

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Book Synopsis Partial Differential Equations VII by : M.A. Shubin

Download or read book Partial Differential Equations VII written by M.A. Shubin and published by Springer. This book was released on 2012-12-22 with total page 274 pages. Available in PDF, EPUB and Kindle. Book excerpt: This EMS volume contains a survey of the principles and advanced techniques of the spectral theory of linear differential and pseudodifferential operators in finite-dimensional spaces. Also including a special section of Sunada's recent solution of Kac's celebrated problem of whether or not "one can hear the shape of a drum".

The Weyl Operator and its Generalization

Author: Leon Cohen

Publisher: Springer Science & Business Media

Published: 2012-12-13

Total Pages: 167

ISBN-13: 3034802943

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The discovery of quantum mechanics in the years 1925-1930 necessitated the consideration of associating ordinary functions with non-commuting operators. Methods were proposed by Born/Jordan, Kirkwood, and Weyl. Sometime later, Moyal saw the connection between the Weyl rule and the Wigner distribution, which had been proposed by Wigner in 1932 as a way of doing quantum statistical mechanics. The basic idea of associating functions with operators has since been generalized and developed to a high degree. It has found several application fields, including quantum mechanics, pseudo-differential operators, time-frequency analysis, quantum optics, wave propagation, differential equations, image processing, radar, and sonar. This book aims at bringing together the results from the above mentioned fields in a unified manner and showing the reader how the methods have been applied. A wide audience is addressed, particularly students and researchers who want to obtain an up-to-date working knowledge of the field. The mathematics is accessible to the uninitiated reader and is presented in a straightforward manner.

Book Synopsis The Weyl Operator and its Generalization by : Leon Cohen

Download or read book The Weyl Operator and its Generalization written by Leon Cohen and published by Springer Science & Business Media. This book was released on 2012-12-13 with total page 167 pages. Available in PDF, EPUB and Kindle. Book excerpt: The discovery of quantum mechanics in the years 1925-1930 necessitated the consideration of associating ordinary functions with non-commuting operators. Methods were proposed by Born/Jordan, Kirkwood, and Weyl. Sometime later, Moyal saw the connection between the Weyl rule and the Wigner distribution, which had been proposed by Wigner in 1932 as a way of doing quantum statistical mechanics. The basic idea of associating functions with operators has since been generalized and developed to a high degree. It has found several application fields, including quantum mechanics, pseudo-differential operators, time-frequency analysis, quantum optics, wave propagation, differential equations, image processing, radar, and sonar. This book aims at bringing together the results from the above mentioned fields in a unified manner and showing the reader how the methods have been applied. A wide audience is addressed, particularly students and researchers who want to obtain an up-to-date working knowledge of the field. The mathematics is accessible to the uninitiated reader and is presented in a straightforward manner.

Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians

Author: Francis Nier

Publisher: Springer

Published: 2005-01-17

Total Pages: 209

ISBN-13: 3540315535

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There has recently been a renewal of interest in Fokker-Planck operators, motivated by problems in statistical physics, in kinetic equations, and differential geometry. Compared to more standard problems in the spectral theory of partial differential operators, those operators are not self-adjoint and only hypoelliptic. The aim of the analysis is to give, as generally as possible, an accurate qualitative and quantitative description of the exponential return to the thermodynamical equilibrium. While exploring and improving recent results in this direction, this volume proposes a review of known techniques on: the hypoellipticity of polynomial of vector fields and its global counterpart, the global Weyl-Hörmander pseudo-differential calculus, the spectral theory of non-self-adjoint operators, the semi-classical analysis of Schrödinger-type operators, the Witten complexes, and the Morse inequalities.

Book Synopsis Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians by : Francis Nier

Download or read book Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians written by Francis Nier and published by Springer. This book was released on 2005-01-17 with total page 209 pages. Available in PDF, EPUB and Kindle. Book excerpt: There has recently been a renewal of interest in Fokker-Planck operators, motivated by problems in statistical physics, in kinetic equations, and differential geometry. Compared to more standard problems in the spectral theory of partial differential operators, those operators are not self-adjoint and only hypoelliptic. The aim of the analysis is to give, as generally as possible, an accurate qualitative and quantitative description of the exponential return to the thermodynamical equilibrium. While exploring and improving recent results in this direction, this volume proposes a review of known techniques on: the hypoellipticity of polynomial of vector fields and its global counterpart, the global Weyl-Hörmander pseudo-differential calculus, the spectral theory of non-self-adjoint operators, the semi-classical analysis of Schrödinger-type operators, the Witten complexes, and the Morse inequalities.