Pseudodifferential Operators with Automorphic Symbols
Pseudodifferential Operators with Automorphic Symbols

Author: André Unterberger

Publisher: Birkhäuser

Published: 2015-06-22

Total Pages: 208

ISBN-13: 3319186574

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The main results of this book combine pseudo differential analysis with modular form theory. The methods rely for the most part on explicit spectral theory and the extended use of special functions. The starting point is a notion of modular distribution in the plane, which will be new to most readers and relates under the Radon transformation to the classical one of modular form of the non-holomorphic type. Modular forms of the holomorphic type are addressed too in a more concise way, within a general scheme dealing with quantization theory and elementary, but novel, representation-theoretic concepts.

Recent Trends in Toeplitz and Pseudodifferential Operators
Recent Trends in Toeplitz and Pseudodifferential Operators

Author: Roland V. Duduchava

Publisher: Springer Science & Business Media

Published: 2011-02-04

Total Pages: 275

ISBN-13: 303460548X

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The aim of the book is to present new results in operator theory and its applications. In particular, the book is devoted to operators with automorphic symbols, applications of the methods of modern operator theory and differential geometry to some problems of theory of elasticity, quantum mechanics, hyperbolic systems of partial differential equations with multiple characteristics, Laplace-Beltrami operators on manifolds with singular points. Moreover, the book comprises new results in the theory of Wiener-Hopf operators with oscillating symbols, large hermitian Toeplitz band matrices, commutative algebras of Toeplitz operators, and discusses a number of other topics.

Automorphic Pseudodifferential Analysis and Higher Level Weyl Calculi
Automorphic Pseudodifferential Analysis and Higher Level Weyl Calculi

Author: André Unterberger

Publisher: Birkhäuser

Published: 2012-12-06

Total Pages: 250

ISBN-13: 3034879784

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Award-winning monograph of the Ferran Sunyer i Balaguer Prize 2002. The subject of this book is the study of automorphic distributions, by which is meant distributions on R2 invariant under the linear action of SL(2,Z), and of the operators associated with such distributions under the Weyl rule of symbolic calculus. Researchers and postgraduates interested in pseudodifferential analyis, the theory of non-holomorphic modular forms, and symbolic calculi will benefit from the clear exposition and new results and insights.

Operator Algebras, Toeplitz Operators and Related Topics
Operator Algebras, Toeplitz Operators and Related Topics

Author: Wolfram Bauer

Publisher: Springer Nature

Published: 2020-09-01

Total Pages: 467

ISBN-13: 3030446514

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This book features a collection of up-to-date research papers that study various aspects of general operator algebra theory and concrete classes of operators, including a range of applications. Most of the papers included were presented at the International Workshop on Operator Algebras, Toeplitz Operators, and Related Topics, in Boca del Rio, Veracruz, Mexico, in November 2018. The conference, which was attended by more than 30 leading experts in the field, was held in celebration of Nikolai Vasilevski’s 70th birthday, and the contributions are dedicated to him.

Pseudodifferential Analysis, Automorphic Distributions in the Plane and Modular Forms
Pseudodifferential Analysis, Automorphic Distributions in the Plane and Modular Forms

Author: André Unterberger

Publisher: Springer Science & Business Media

Published: 2011-08-06

Total Pages: 305

ISBN-13: 3034801661

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Pseudodifferential analysis, introduced in this book in a way adapted to the needs of number theorists, relates automorphic function theory in the hyperbolic half-plane Π to automorphic distribution theory in the plane. Spectral-theoretic questions are discussed in one or the other environment: in the latter one, the problem of decomposing automorphic functions in Π according to the spectral decomposition of the modular Laplacian gives way to the simpler one of decomposing automorphic distributions in R2 into homogeneous components. The Poincaré summation process, which consists in building automorphic distributions as series of g-transforms, for g E SL(2;Z), of some initial function, say in S(R2), is analyzed in detail. On Π, a large class of new automorphic functions or measures is built in the same way: one of its features lies in an interpretation, as a spectral density, of the restriction of the zeta function to any line within the critical strip. The book is addressed to a wide audience of advanced graduate students and researchers working in analytic number theory or pseudo-differential analysis.

Pseudodifferential Methods in Number Theory
Pseudodifferential Methods in Number Theory

Author: André Unterberger

Publisher: Birkhäuser

Published: 2018-07-16

Total Pages: 175

ISBN-13: 3319927078

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Classically developed as a tool for partial differential equations, the analysis of operators known as pseudodifferential analysis is here regarded as a possible help in questions of arithmetic. The operators which make up the main subject of the book can be characterized in terms of congruence arithmetic. They enjoy a Eulerian structure, and are applied to the search for new conditions equivalent to the Riemann hypothesis. These consist in the validity of certain parameter-dependent estimates for a class of Hermitian forms of finite rank. The Littlewood criterion, involving sums of Möbius coefficients, and the Weil so-called explicit formula, which leads to his positivity criterion, fit within this scheme, using in the first case Weyl's pseudodifferential calculus, in the second case Fuchs'. The book should be of interest to people looking for new possible approaches to the Riemann hypothesis, also to new perspectives on pseudodifferential analysis and on the way it combines with modular form theory. Analysts will have no difficulty with the arithmetic aspects, with which, save for very few exceptions, no previous acquaintance is necessary.

Handbook of Analytic Operator Theory
Handbook of Analytic Operator Theory

Author: Kehe Zhu

Publisher: CRC Press

Published: 2019-05-10

Total Pages: 360

ISBN-13: 1351045547

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Handbook of Analytic Operator Theory thoroughly covers the subject of holomorphic function spaces and operators acting on them. The spaces covered include Bergman spaces, Hardy spaces, Fock spaces and the Drury-Averson space. Operators discussed in the book include Toeplitz operators, Hankel operators, composition operators, and Cowen-Douglas class operators. The volume consists of eleven articles in the general area of analytic function spaces and operators on them. Each contributor focuses on one particular topic, for example, operator theory on the Drury-Aversson space, and presents the material in the form of a survey paper which contains all the major results in the area and includes all relevant references. The overalp between this volume and existing books in the area is minimal. The material on two-variable weighted shifts by Curto, the Drury-Averson space by Fang and Xia, the Cowen-Douglas class by Misra, and operator theory on the bi-disk by Yang has never appeared in book form before. Features: The editor of the handbook is a widely known and published researcher on this topic The handbook's contributors are a who's=who of top researchers in the area The first contributed volume on these diverse topics

Partial Differential Equations VIII
Partial Differential Equations VIII

Author: M.A. Shubin

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 266

ISBN-13: 3642489443

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This volume of the EMS contains three articles, on linear overdetermined systems of partial differential equations, dissipative Schroedinger operators, and index theorems. Each article presents a comprehensive survey of its subject, discussing fundamental results such as the construction of compatibility operators and complexes for elliptic, parabolic and hyperbolic coercive problems, the method of functional models and the Atiyah-Singer index theorem and its generalisations. Both classical and recent results are explained in detail and illustrated by means of examples.

Quantization and Non-holomorphic Modular Forms
Quantization and Non-holomorphic Modular Forms

Author: André Unterberger

Publisher: Springer

Published: 2007-05-06

Total Pages: 251

ISBN-13: 3540446605

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This is a new approach to the theory of non-holomorphic modular forms, based on ideas from quantization theory or pseudodifferential analysis. Extending the Rankin-Selberg method so as to apply it to the calculation of the Roelcke-Selberg decomposition of the product of two Eisenstein series, one lets Maass cusp-forms appear as residues of simple, Eisenstein-like, series. Other results, based on quantization theory, include a reinterpretation of the Lax-Phillips scattering theory for the automorphic wave equation, in terms of distributions on R2 automorphic with respect to the linear action of SL(2,Z).

Jacobi-Like Forms, Pseudodifferential Operators, and Quasimodular Forms
Jacobi-Like Forms, Pseudodifferential Operators, and Quasimodular Forms

Author: YoungJu Choie

Publisher: Springer Nature

Published: 2019-11-20

Total Pages: 307

ISBN-13: 3030291235

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This book explores various properties of quasimodular forms, especially their connections with Jacobi-like forms and automorphic pseudodifferential operators. The material that is essential to the subject is presented in sufficient detail, including necessary background on pseudodifferential operators, Lie algebras, etc., to make it accessible also to non-specialists. The book also covers a sufficiently broad range of illustrations of how the main themes of the book have occurred in various parts of mathematics to make it attractive to a wider audience. The book is intended for researchers and graduate students in number theory.