Spectral Theory Of Infinite Area Hyperbolic Surfaces

Author : David Borthwick
Publisher : Birkhäuser
Page : 471 pages
File Size : 54,6 Mb
Release : 2016-07-12
Category : Mathematics
ISBN : 9783319338774

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Spectral Theory of Infinite-Area Hyperbolic Surfaces by David Borthwick Pdf

This text introduces geometric spectral theory in the context of infinite-area Riemann surfaces, providing a comprehensive account of the most recent developments in the field. For the second edition the context has been extended to general surfaces with hyperbolic ends, which provides a natural setting for development of the spectral theory while still keeping technical difficulties to a minimum. All of the material from the first edition is included and updated, and new sections have been added. Topics covered include an introduction to the geometry of hyperbolic surfaces, analysis of the resolvent of the Laplacian, scattering theory, resonances and scattering poles, the Selberg zeta function, the Poisson formula, distribution of resonances, the inverse scattering problem, Patterson-Sullivan theory, and the dynamical approach to the zeta function. The new sections cover the latest developments in the field, including the spectral gap, resonance asymptotics near the critical line, and sharp geometric constants for resonance bounds. A new chapter introduces recently developed techniques for resonance calculation that illuminate the existing results and conjectures on resonance distribution. The spectral theory of hyperbolic surfaces is a point of intersection for a great variety of areas, including quantum physics, discrete groups, differential geometry, number theory, complex analysis, and ergodic theory. This book will serve as a valuable resource for graduate students and researchers from these and other related fields. Review of the first edition: "The exposition is very clear and thorough, and essentially self-contained; the proofs are detailed...The book gathers together some material which is not always easily available in the literature...To conclude, the book is certainly at a level accessible to graduate students and researchers from a rather large range of fields. Clearly, the reader...would certainly benefit greatly from it." (Colin Guillarmou, Mathematical Reviews, Issue 2008 h)

Author : David Borthwick
Publisher : Springer Nature
Page : 339 pages
File Size : 52,6 Mb
Release : 2020-03-12
Category : Mathematics
ISBN : 9783030380021

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Spectral Theory by David Borthwick Pdf

This textbook offers a concise introduction to spectral theory, designed for newcomers to functional analysis. Curating the content carefully, the author builds to a proof of the spectral theorem in the early part of the book. Subsequent chapters illustrate a variety of application areas, exploring key examples in detail. Readers looking to delve further into specialized topics will find ample references to classic and recent literature. Beginning with a brief introduction to functional analysis, the text focuses on unbounded operators and separable Hilbert spaces as the essential tools needed for the subsequent theory. A thorough discussion of the concepts of spectrum and resolvent follows, leading to a complete proof of the spectral theorem for unbounded self-adjoint operators. Applications of spectral theory to differential operators comprise the remaining four chapters. These chapters introduce the Dirichlet Laplacian operator, Schrödinger operators, operators on graphs, and the spectral theory of Riemannian manifolds. Spectral Theory offers a uniquely accessible introduction to ideas that invite further study in any number of different directions. A background in real and complex analysis is assumed; the author presents the requisite tools from functional analysis within the text. This introductory treatment would suit a functional analysis course intended as a pathway to linear PDE theory. Independent later chapters allow for flexibility in selecting applications to suit specific interests within a one-semester course.

Author : Alex Barnett
Publisher : American Mathematical Soc.
Page : 339 pages
File Size : 54,7 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821853191

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Spectral Geometry by Alex Barnett Pdf

This volume contains the proceedings of the International Conference on Spectral Geometry, held July 19-23, 2010, at Dartmouth College, Dartmouth, New Hampshire. Eigenvalue problems involving the Laplace operator on manifolds have proven to be a consistently fertile area of geometric analysis with deep connections to number theory, physics, and applied mathematics. Key questions include the measures to which eigenfunctions of the Laplacian on a Riemannian manifold condense in the limit of large eigenvalue, and the extent to which the eigenvalues and eigenfunctions of a manifold encode its geometry. In this volume, research and expository articles, including those of the plenary speakers Peter Sarnak and Victor Guillemin, address the flurry of recent progress in such areas as quantum unique ergodicity, isospectrality, semiclassical measures, the geometry of nodal lines of eigenfunctions, methods of numerical computation, and spectra of quantum graphs. This volume also contains mini-courses on spectral theory for hyperbolic surfaces, semiclassical analysis, and orbifold spectral geometry that prepared the participants, especially graduate students and young researchers, for conference lectures.

Author : Jens Bölte,Frank Steiner
Publisher : Unknown
Page : 272 pages
File Size : 55,9 Mb
Release : 2012
Category : Cosmology
ISBN : 1139887734

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Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology by Jens Bölte,Frank Steiner Pdf

Hyperbolic geometry is a classical subject in pure mathematics which has exciting applications in theoretical physics. In this book leading experts introduce hyperbolic geometry and Maass waveforms and discuss applications in quantum chaos and cosmology. The book begins with an introductory chapter detailing the geometry of hyperbolic surfaces and includes numerous worked examples and exercises to give the reader a solid foundation for the rest of the book. In later chapters the classical version of Selberg's trace formula is derived in detail and transfer operators are developed as tools in the spectral theory of Laplace-Beltrami operators on modular surfaces. The computation of Maass waveforms and associated eigenvalues of the hyperbolic Laplacian on hyperbolic manifolds are also presented in a comprehensive way. This book will be valuable to graduate students and young researchers, as well as for those experienced scientists who want a detailed exposition of the subject.

Author : Plamen Stefanov,András Vasy,Maciej Zworski
Publisher : American Mathematical Soc.
Page : 309 pages
File Size : 51,5 Mb
Release : 2014-05-05
Category : Mathematics
ISBN : 9781470410797

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Inverse Problems and Applications by Plamen Stefanov,András Vasy,Maciej Zworski Pdf

This volume contains the proceedings of two conferences on Inverse Problems and Applications, held in 2012, to celebrate the work of Gunther Uhlmann. The first conference was held at the University of California, Irvine, from June 18-22, 2012, and the second was held at Zhejiang University, Hangzhou, China, from September 17-21, 2012. The topics covered include inverse problems in medical imaging, scattering theory, geometry and image processing, and the mathematical theory of cloaking, as well as methods related to inverse problems.

Author : Fritz Gesztesy
Publisher : American Mathematical Soc.
Page : 528 pages
File Size : 48,9 Mb
Release : 2007
Category : Mathematical physics
ISBN : 9780821842485

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Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon's 60th Birthday by Fritz Gesztesy Pdf

This Festschrift had its origins in a conference called SimonFest held at Caltech, March 27-31, 2006, to honor Barry Simon's 60th birthday. It is not a proceedings volume in the usual sense since the emphasis of the majority of the contributions is on reviews of the state of the art of certain fields, with particular focus on recent developments and open problems. The bulk of the articles in this Festschrift are of this survey form, and a few review Simon's contributions to aparticular area. Part 1 contains surveys in the areas of Quantum Field Theory, Statistical Mechanics, Nonrelativistic Two-Body and $N$-Body Quantum Systems, Resonances, Quantum Mechanics with Electric and Magnetic Fields, and the Semiclassical Limit. Part 2 contains surveys in the areas of Random andErgodic Schrodinger Operators, Singular Continuous Spectrum, Orthogonal Polynomials, and Inverse Spectral Theory. In several cases, this collection of surveys portrays both the history of a subject and its current state of the art. A substantial part of the contributions to this Festschrift are survey articles on the state of the art of certain areas with special emphasis on open problems. This will benefit graduate students as well as researchers who want to get a quick, yet comprehensiveintroduction into an area covered in this volume.

Author : Joel S. Feldman,Richard Gerd Froese,Lon M. Rosen
Publisher : American Mathematical Soc.
Page : 316 pages
File Size : 49,7 Mb
Release : 1995
Category : Science
ISBN : 0821870491

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Mathematical Quantum Theory II by Joel S. Feldman,Richard Gerd Froese,Lon M. Rosen Pdf

Author : Charles L. Epstein
Publisher : American Mathematical Soc.
Page : 174 pages
File Size : 48,7 Mb
Release : 1985
Category : Spectral theory
ISBN : 9780821823361

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The Spectral Theory of Geometrically Periodic Hyperbolic 3-Manifolds by Charles L. Epstein Pdf

In this paper we develop the spectral theory of the Laplace-Beltrami operator for geometrically periodic hyperbolic 3-manifolds, [double-struck capital]H3/G. Using the theory of holomorphic families of operators, we obtain a quantitative description of the absolutely continuous spectrum.

Author : Dennis A. Hejhal
Publisher : American Mathematical Soc.
Page : 146 pages
File Size : 53,8 Mb
Release : 1990
Category : Mathematics
ISBN : 9780821824993

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Regular $b$-Groups, Degenerating Riemann Surfaces, and Spectral Theory by Dennis A. Hejhal Pdf

This paper is concerned with the spectral theory of the Laplacian as the underlying Riemann surface is "pinched down" to a surface with nodes. The problem is attacked from the (general) standpoint of regular b-groups and the Selberg trace formula.

Author : Józef Dodziuk,Jay Jorgenson
Publisher : American Mathematical Soc.
Page : 90 pages
File Size : 42,5 Mb
Release : 1998
Category : Asymptotic expansions
ISBN : 9780821808375

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Spectral Asymptotics on Degenerating Hyperbolic 3-Manifolds by Józef Dodziuk,Jay Jorgenson Pdf

In this volume, the authors study asymptotics of the geometry and spectral theory of degenerating sequences of finite volume hyperbolic manifolds of three dimensions. Thurston's hyperbolic surgery theorem assets the existence of non-trivial sequences of finite volume hyperbolic three manifolds which converge to a three manifold with additional cusps. In the geometric aspect of their study, the authors use the convergence of hyperbolic metrics on the thick parts of the manifolds under consideration to investigate convergentce of tubes in the manifolds of the sequence to cusps of the limiting manifold. In the specral theory aspect of the work, they prove convergence of heat kernels. They then define a regualrized heat race associated to any finite volume, complete, hyperbolic three manifold, and study its asymptotic behaviour through degeneration. As an application of the analysis of the regularized heat trace, they study asymptotic behaviours of the spectral zeta function, determinant of the Laplacian, Selberg zeta function, and spectral counting functions through degeneration. The authors' methods are an adaptation to three dimensions of the earlier work of Jorgenson and Lundelius who investigated the asymptotic behaviour of spectral functions on degenerating families of finite area hyperbolic Riemann surfaces.

Author : Semyon Dyatlov,Maciej Zworski
Publisher : American Mathematical Soc.
Page : 634 pages
File Size : 52,5 Mb
Release : 2019-09-10
Category : Frequencies of oscillating systems
ISBN : 9781470443665

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Mathematical Theory of Scattering Resonances by Semyon Dyatlov,Maciej Zworski Pdf

Scattering resonances generalize bound states/eigenvalues for systems in which energy can scatter to infinity. A typical resonance has a rate of oscillation (just as a bound state does) and a rate of decay. Although the notion is intrinsically dynamical, an elegant mathematical formulation comes from considering meromorphic continuations of Green's functions. The poles of these meromorphic continuations capture physical information by identifying the rate of oscillation with the real part of a pole and the rate of decay with its imaginary part. An example from mathematics is given by the zeros of the Riemann zeta function: they are, essentially, the resonances of the Laplacian on the modular surface. The Riemann hypothesis then states that the decay rates for the modular surface are all either or . An example from physics is given by quasi-normal modes of black holes which appear in long-time asymptotics of gravitational waves. This book concentrates mostly on the simplest case of scattering by compactly supported potentials but provides pointers to modern literature where more general cases are studied. It also presents a recent approach to the study of resonances on asymptotically hyperbolic manifolds. The last two chapters are devoted to semiclassical methods in the study of resonances.

Author : Kurt O. Friedrichs
Publisher : Springer Science & Business Media
Page : 253 pages
File Size : 45,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461263968

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Spectral Theory of Operators in Hilbert Space by Kurt O. Friedrichs Pdf

The present lectures intend to provide an introduction to the spectral analysis of self-adjoint operators within the framework of Hilbert space theory. The guiding notion in this approach is that of spectral representation. At the same time the notion of function of an operator is emphasized. The formal aspects of these concepts are explained in the first two chapters. Only then is the notion of Hilbert space introduced. The following three chapters concern bounded, completely continuous, and non-bounded operators. Next, simple differential operators are treated as operators in Hilbert space, and the final chapter deals with the perturbation of discrete and continuous spectra. The preparation of the original version of these lecture notes was greatly helped by the assistance of P. Rejto. Various valuable suggestions made by him and by R. Lewis have been incorporated. The present version of the notes contains extensive modifica tions, in particular in the chapters on bounded and unbounded operators. February, 1973 K.O.F. PREFACE TO THE SECOND PRINTING The second printing (1980) is a basically unchanged reprint in which a number of minor errors were corrected. The author wishes to thank Klaus Schmidt (Lausanne) and John Sylvester (New York) for their lists of errors. v TABLE OF CONTENTS I. Spectral Representation 1 1. Three typical problems 1 12 2. Linear space and functional representation.

Author : K. B. Laursen,Michael Neumann
Publisher : Oxford University Press
Page : 610 pages
File Size : 51,5 Mb
Release : 2000
Category : Mathematics
ISBN : 0198523815

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An Introduction to Local Spectral Theory by K. B. Laursen,Michael Neumann Pdf

Modern local spectral theory is built on the classical spectral theorem, a fundamental result in single-operator theory and Hilbert spaces. This book provides an in-depth introduction to the natural expansion of this fascinating topic of Banach space operator theory. It gives complete coverage of the field, including the fundamental recent work by Albrecht and Eschmeier which provides the full duality theory for Banach space operators. One of its highlights are the many characterizations of decomposable operators, and of other related, important classes of operators, including identifications of distinguished parts, and results on permanence properties of spectra with respect to several types of similarity. Written in a careful and detailed style, it contains numerous examples, many simplified proofs of classical results, extensive references, and open problems, suitable for continued research.

Author : Gilbert Helmberg
Publisher : Courier Dover Publications
Page : 370 pages
File Size : 48,5 Mb
Release : 2008-06-11
Category : Science
ISBN : 9780486466224

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Introduction to Spectral Theory in Hilbert Space by Gilbert Helmberg Pdf

This introduction to Hilbert space, bounded self-adjoint operators, the spectrum of an operator, and operators' spectral decomposition is accessible to readers familiar with analysis and analytic geometry. 1969 edition.

Author : William Arveson
Publisher : Springer Science & Business Media
Page : 143 pages
File Size : 55,6 Mb
Release : 2006-04-18
Category : Mathematics
ISBN : 9780387215181

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A Short Course on Spectral Theory by William Arveson Pdf

This book presents the basic tools of modern analysis within the context of the fundamental problem of operator theory: to calculate spectra of specific operators on infinite dimensional spaces, especially operators on Hilbert spaces. The tools are diverse, and they provide the basis for more refined methods that allow one to approach problems that go well beyond the computation of spectra: the mathematical foundations of quantum physics, noncommutative K-theory, and the classification of simple C*-algebras being three areas of current research activity which require mastery of the material presented here.