• Mathematics

Microlocal Analysis for Differential Operators

An Introduction
Author: Alain Grigis,Johannes Sjöstrand
Publisher: Cambridge University Press
ISBN: 9780521449861
Category: Mathematics
Page: 151
View: 5334
This book corresponds to a graduate course given many times by the authors, and should prove to be useful to mathematicians and theoretical physicists.

    • Mathematics

Partial Differential Equations IV

Microlocal Analysis and Hyperbolic Equations
Author: Yu.V. Egorov,M.A. Shubin
Publisher: Springer Science & Business Media
ISBN: 3662092077
Category: Mathematics
Page: 244
View: 7860
A two-part monograph covering recent research in an important field, previously scattered in numerous journals, including the latest results in the theory of mixed problems for hyperbolic operators. The book is hence of immense value to graduate students and researchers in partial differential equations and theoretical physics.

    • Mathematics

An Introduction to Semiclassical and Microlocal Analysis


Author: Andre Martinez
Publisher: Springer Science & Business Media
ISBN: 9780387953441
Category: Mathematics
Page: 190
View: 7138
"Various exercises illustrate the chief results of each chapter while introducing the reader to further developments of the theory. Applications to the study of the Schrodinger operator are also discussed, to further the understanding of new notions or general results by placing them in the context of quantum mechanics. This book is aimed at nonspecialists of the subject, and the only required prerequisite is a basic knowledge of the theory of distributions."--BOOK JACKET.


    • Mathematics

Lectures on Nonlinear Hyperbolic Differential Equations


Author: Lars Hörmander
Publisher: Springer Science & Business Media
ISBN: 9783540629214
Category: Mathematics
Page: 289
View: 4035
In this introductory textbook, a revised and extended version of well-known lectures by L. Hörmander from 1986, four chapters are devoted to weak solutions of systems of conservation laws. Apart from that the book only studies classical solutions. Two chapters concern the existence of global solutions or estimates of the lifespan for solutions of nonlinear perturbations of the wave or Klein-Gordon equation with small initial data. Four chapters are devoted to microanalysis of the singularities of the solutions. This part assumes some familiarity with pseudodifferential operators which are standard in the theory of linear differential operators, but the extension to the more exotic classes of opertors needed in the nonlinear theory is presented in complete detail.

    • Mathematics

Seminar on Micro-Local Analysis. (AM-93)


Author: Victor Guillemin,Masaki Kashiwara,Takahiro Kawai
Publisher: Princeton University Press
ISBN: 1400881579
Category: Mathematics
Page: 152
View: 4775
Based on a seminar sponsored by the Institute for Advanced Study in 1977-1978, this set of papers introduces micro-local analysis concisely and clearly to mathematicians with an analytical background. The papers treat the theory of microfunctions and applications such as boundary values of elliptic partial differential equations, propagation of singularities in the vicinity of degenerate characteristics, holonomic systems, Feynman integrals from the hyperfunction point of view, and harmonic analysis on Lie groups.

    • Mathematics

Microlocal Analysis and Spectral Theory


Author: Luigi Rodino
Publisher: Springer Science & Business Media
ISBN: 9401156263
Category: Mathematics
Page: 444
View: 2041
The NATO Advanced Study Institute "Microlocal Analysis and Spectral The ory" was held in Tuscany (Italy) at Castelvecchio Pascoli, in the district of Lucca, hosted by the international vacation center "11 Ciocco" , from September 23 to October 3, 1996. The Institute recorded the considerable progress realized recently in the field of Microlocal Analysis. In a broad sense, Microlocal Analysis is the modern version of the classical Fourier technique in solving partial differential equa tions, where now the localization proceeding takes place with respect to the dual variables too. Precisely, through the tools of pseudo-differential operators, wave-front sets and Fourier integral operators, the general theory of the lin ear partial differential equations is now reaching a mature form, in the frame of Schwartz distributions or other generalized functions. At the same time, Microlocal Analysis has grown up into a definite and independent part of Math ematical Analysis, with other applications all around Mathematics and Physics, one major theme being Spectral Theory for Schrodinger equation in Quantum Mechanics.

    • Mathematics

Semiclassical Analysis


Author: Maciej Zworski
Publisher: American Mathematical Soc.
ISBN: 0821883208
Category: Mathematics
Page: 431
View: 4911
This book is an excellent, comprehensive introduction to semiclassical analysis. I believe it will become a standard reference for the subject. --Alejandro Uribe, University of Michigan Semiclassical analysis provides PDE techniques based on the classical-quantum (particle-wave) correspondence. These techniques include such well-known tools as geometric optics and the Wentzel-Kramers-Brillouin approximation. Examples of problems studied in this subject are high energy eigenvalue asymptotics and effective dynamics for solutions of evolution equations. From the mathematical point of view, semiclassical analysis is a branch of microlocal analysis which, broadly speaking, applies harmonic analysis and symplectic geometry to the study of linear and nonlinear PDE. The book is intended to be a graduate level text introducing readers to semiclassical and microlocal methods in PDE. It is augmented in later chapters with many specialized advanced topics which provide a link to current research literature.

    • Mathematics

Spectral Asymptotics in the Semi-Classical Limit


Author: M. Dimassi,J. Sjostrand
Publisher: Cambridge University Press
ISBN: 9780521665445
Category: Mathematics
Page: 227
View: 2484
This book presents the basic methods and applications in semiclassical approximation in the light of developments.

    • Mathematics

Microlocal Analysis and Precise Spectral Asymptotics


Author: Victor Ivrii
Publisher: Springer Science & Business Media
ISBN: 3662124963
Category: Mathematics
Page: 733
View: 990
The problem of spectral asymptotics, in particular the problem of the asymptotic dis tribution 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.

    • Mathematics

Pseudo-differential Operators and the Nash-Moser Theorem


Author: Serge Alinhac,Patrick Gérard
Publisher: American Mathematical Soc.
ISBN: 0821834541
Category: Mathematics
Page: 168
View: 4414
This book presents two essential and apparently unrelated subjects. The first, microlocal analysis and the theory of pseudo-differential operators, is a basic tool in the study of partial differential equations and in analysis on manifolds. The second, the Nash-Moser theorem, continues to be fundamentally important in geometry, dynamical systems and nonlinear PDE. Each of the subjects, which are of interest in their own right as well as for applications, can be learned separately. But the book shows the deep connections between the two themes, particularly in the middle part, which is devoted to Littlewood-Paley theory, dyadic analysis, and the paradifferential calculus and its application to interpolation inequalities. An important feature is the elementary and self-contained character of the text, to which many exercises and an introductory Chapter $0$ with basic material have been added. This makes the book readable by graduate students or researchers from one subject who are interested in becoming familiar with the other. It can also be used as a textbook for a graduate course on nonlinear PDE or geometry.

    • Mathematics

Pseudodifferential Operators and Spectral Theory


Author: M.A. Shubin
Publisher: Springer Science & Business Media
ISBN: 3642565794
Category: Mathematics
Page: 288
View: 4071
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.

    • Mathematics

Spectral Theory, Microlocal Analysis, Singular Manifolds

Advances in Partial Differential Equations
Author: Michael Demuth,Elmar Schrohe,Bert-Wolfgang Schulze,Johannes Sjöstrand
Publisher: Wiley-VCH
ISBN: 9783527401208
Category: Mathematics
Page: 368
View: 1386
The spectral theory of differential operators is a challenging subject with deep connections to many branches of mathematics and mathematical physics. It is a central issue in this volume of Advances in Partial Differential Equations The first contribution addresses domain perturbations for generalized Schr?dinger operators and the influence of the capacity on spectral data. There follows an article discussing the minimal smoothness assumptions on the domain under which the asymptotics of the counting function for the eigenvalues of elliptic boundary value problems can be determined. Systems of h-pseudo-differential operators on the half-line are studied in the next paper. The results concern existence and distribution of resonances for various semi-classical regimes. Three further articles are devoted to the regularity and symptotics of solutions to partial differential equations on singular manifolds. A very efficient tool is the combination of suitable operator algebras and pseudo-differential calculi with sufficiently rich symbolic structures. One paper considers the case of manifolds with non-compact ends, another the case of higher cuspidal singularities. A final contribution treats degenerate hyperbolic equations.

    • Mathematics

Algebraic Analysis of Differential Equations

from Microlocal Analysis to Exponential Asymptotics
Author: T. Aoki,H. Majima,Y. Takei,N. Tose
Publisher: Springer Science & Business Media
ISBN: 9784431732402
Category: Mathematics
Page: 352
View: 765
This volume contains 23 articles on algebraic analysis of differential equations and related topics, most of which were presented as papers at the conference "Algebraic Analysis of Differential Equations – from Microlocal Analysis to Exponential Asymptotics" at Kyoto University in 2005. This volume is dedicated to Professor Takahiro Kawai, who is one of the creators of microlocal analysis and who introduced the technique of microlocal analysis into exponential asymptotics.

    • MATHEMATICS

Tools for PDE

Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials
Author: Michael Eugene Taylor
Publisher: American Mathematical Soc.
ISBN: 0821843788
Category: MATHEMATICS
Page: 257
View: 9587
This book develops three related tools that are useful in the analysis of partial differential equations (PDEs), arising from the classical study of singular integral operators: pseudodifferential operators, paradifferential operators, and layer potentials. A theme running throughout the work is the treatment of PDE in the presence of relatively little regularity. The first chapter studies classes of pseudodifferential operators whose symbols have a limited degree of regularity; the second chapter shows how paradifferential operators yield sharp estimates on the action of various nonlinear operators on function spaces. The third chapter applies this material to an assortment of results in PDE, including regularity results for elliptic PDE with rough coefficients, planar fluid flows on rough domains, estimates on Riemannian manifolds given weak bounds on Ricci tensor, div-curl estimates, and results on propagation of singularities for wave equations with rough coefficients. The last chapter studies the method of layer potentials on Lipschitz domains, concentrating on applications to boundary problems for elliptic PDE with variable coefficients.

    • Mathematics

Elementary Introduction to the Theory of Pseudodifferential Operators


Author: Xavier Saint Raymond
Publisher: Routledge
ISBN: 1351452924
Category: Mathematics
Page: 120
View: 3906
In the 19th century, the Fourier transformation was introduced to study various problems of partial differential equations. Since 1960, this old tool has been developed into a well-organized theory called microlocal analysis that is based on the concept of the pseudo-differential operator. This book provides the fundamental knowledge non-specialists need in order to use microlocal analysis. It is strictly mathematical in the sense that it contains precise definitions, statements of theorems and complete proofs, and follows the usual method of pure mathematics. The book explains the origin of the theory (i.e., Fourier transformation), presents an elementary construcion of distribution theory, and features a careful exposition of standard pseudodifferential theory. Exercises, historical notes, and bibliographical references are included to round out this essential book for mathematics students; engineers, physicists, and mathematicians who use partial differential equations; and advanced mathematics instructors.

    • Mathematics

Lectures on Linear Partial Differential Equations


Author: Grigoriĭ Ilʹich Eskin
Publisher: American Mathematical Soc.
ISBN: 0821852841
Category: Mathematics
Page: 410
View: 6026
This book is a reader-friendly, relatively short introduction to the modern theory of linear partial differential equations. An effort has been made to present complete proofs in an accessible and self-contained form. The first three chapters are on elementary distribution theory and Sobolev spaces with many examples and applications to equations with constant coefficients. The following chapters study the Cauchy problem for parabolic and hyperbolic equations, boundary value problems for elliptic equations, heat trace asymptotics, and scattering theory. The book also covers microlocal analysis, including the theory of pseudodifferential and Fourier integral operators, and the propagation of singularities for operators of real principal type. Among the more advanced topics are the global theory of Fourier integral operators and the geometric optics construction in the large, the Atiyah-Singer index theorem in $\mathbb R^n$, and the oblique derivative problem.

    • Mathematics

Evolution Equations


Author: David Ellwood,Igor Rodnianski,Gigliola Staffilani,Jared Wunsch
Publisher: American Mathematical Soc.
ISBN: 0821868616
Category: Mathematics
Page: 572
View: 7452
This volume is a collection of notes from lectures given at the 2008 Clay Mathematics Institute Summer School, held in Zürich, Switzerland. The lectures were designed for graduate students and mathematicians within five years of the Ph.D., and the main focus of the program was on recent progress in the theory of evolution equations. Such equations lie at the heart of many areas of mathematical physics and arise not only in situations with a manifest time evolution (such as linear and nonlinear wave and Schrödinger equations) but also in the high energy or semi-classical limits of elliptic problems. The three main courses focused primarily on microlocal analysis and spectral and scattering theory, the theory of the nonlinear Schrödinger and wave equations, and evolution problems in general relativity. These major topics were supplemented by several mini-courses reporting on the derivation of effective evolution equations from microscopic quantum dynamics; on wave maps with and without symmetries; on quantum N-body scattering, diffraction of waves, and symmetric spaces; and on nonlinear Schrödinger equations at critical regularity. Although highly detailed treatments of some of these topics are now available in the published literature, in this collection the reader can learn the fundamental ideas and tools with a minimum of technical machinery. Moreover, the treatment in this volume emphasizes common themes and techniques in the field, including exact and approximate conservation laws, energy methods, and positive commutator arguments. Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).

    • Mathematics

Phase Space Analysis of Partial Differential Equations


Author: Antonio Bove,Ferruccio Colombini,Daniele Del Santo
Publisher: Springer Science & Business Media
ISBN: 9780817645212
Category: Mathematics
Page: 343
View: 2001
This collection of original articles and surveys treats linear and nonlinear aspects of the theory of partial differential equations. Phase space analysis methods, also known as microlocal analysis, have yielded striking results over the past years and have become one of the main tools of investigation. Equally important is their role in many applications to physics, for example, in quantum and spectral theory. Contributors: H. Bahouri, M. Baouendi, E. Bernardi, M. Bony, A. Bove, N. Burq, J.-Y. Chemin, F. Colombini, T. Colin, P. Cordaro, G. Eskin, X. Fu, N. Hanges, G. Métivier, P. Michor, T. Nishitani, A. Parmeggiani,L. Pernazza, V. Petkov, F. Planchon, M. Prizzi, D. Del Santo, D. Tartakof, D. Tataru, F. Treves, C.-J. Xu, X. Zhang, E. Zuazua

    • Mathematics

Pseudo-Differential Operators: Analysis, Applications and Computations


Author: Luigi Rodino,M. W. Wong,Hongmei Zhu
Publisher: Springer Science & Business Media
ISBN: 9783034800495
Category: Mathematics
Page: 308
View: 1158
This volume consists of eighteen peer-reviewed papers related to lectures on pseudo-differential operators presented at the meeting of the ISAAC Group in Pseudo-Differential Operators (IGPDO) held at Imperial College London on July 13-18, 2009. Featured in this volume are the analysis, applications and computations of pseudo-differential operators in mathematics, physics and signal analysis. This volume is a useful complement to the volumes “Advances in Pseudo-Differential Operators”, “Pseudo-Differential Operators and Related Topics”, “Modern Trends in Pseudo-Differential Operators”, “New Developments in Pseudo-Differential Operators” and “Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations” published in the same series in, respectively, 2004, 2006, 2007, 2009 and 2010.