Download or read book Introduction to spectral theory and inverse problem on asymptotically hyperbolic manifolds written by Hiroshi Isozaki and published by . This book was released on 2014-06 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This manuscript is devoted to a rigorous and detailed exposition of the spectral theory and associated forward and inverse scattering problems for the Laplace-Beltrami operators on asymptotically hyperbolic manifolds. Based upon the classical stationary scattering theory in ℝn, the key point of the approach is the generalized Fourier transform, which serves as the basic tool to introduce and analyse the time-dependent wave operators and the S-matrix. The crucial role is played by the characterization of the space of the scattering solutions for the Helmholtz equations utilizing a properly defined Besov-type space. After developing the scattering theory, we describe, for some cases, the inverse scattering on the asymptotically hyperbolic manifolds by adopting, for the considered case, the boundary control method for inverse problems.The manuscript is aimed at graduate students and young mathematicians interested in spectral and scattering theories, analysis on hyperbolic manifolds and theory of inverse problems. We try to make it self-consistent and, to a large extent, not dependent on the existing treatises on these topics. To our best knowledge, it is the first comprehensive description of these theories in the context of the asymptotically hyperbolic manifolds.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets

Download or read book Inverse Spectral and Scattering Theory written by Hiroshi Isozaki and published by Springer Nature. This book was released on 2020-09-26 with total page 130 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this book is to provide basic knowledge of the inverse problems arising in various areas in mathematics, physics, engineering, and medical science. These practical problems boil down to the mathematical question in which one tries to recover the operator (coefficients) or the domain (manifolds) from spectral data. The characteristic properties of the operators in question are often reduced to those of Schrödinger operators. We start from the 1-dimensional theory to observe the main features of inverse spectral problems and then proceed to multi-dimensions. The first milestone is the Borg–Levinson theorem in the inverse Dirichlet problem in a bounded domain elucidating basic motivation of the inverse problem as well as the difference between 1-dimension and multi-dimension. The main theme is the inverse scattering, in which the spectral data is Heisenberg’s S-matrix defined through the observation of the asymptotic behavior at infinity of solutions. Significant progress has been made in the past 30 years by using the Faddeev–Green function or the complex geometrical optics solution by Sylvester and Uhlmann, which made it possible to reconstruct the potential from the S-matrix of one fixed energy. One can also prove the equivalence of the knowledge of S-matrix and that of the Dirichlet-to-Neumann map for boundary value problems in bounded domains. We apply this idea also to the Dirac equation, the Maxwell equation, and discrete Schrödinger operators on perturbed lattices. Our final topic is the boundary control method introduced by Belishev and Kurylev, which is for the moment the only systematic method for the reconstruction of the Riemannian metric from the boundary observation, which we apply to the inverse scattering on non-compact manifolds. We stress that this book focuses on the lucid exposition of these problems and mathematical backgrounds by explaining the basic knowledge of functional analysis and spectral theory, omitting the technical details in order to make the book accessible to graduate students as an introduction to partial differential equations (PDEs) and functional analysis.

Download or read book Spectral Theory and Partial Differential Equations written by James V Ralston and published by American Mathematical Soc.. This book was released on 2015 with total page 210 pages. Available in PDF, EPUB and Kindle. Book excerpt: Contains the proceedings of the Conference on Spectral Theory and Partial Differential Equations, held in honor of James Ralston's 70th Birthday. Papers cover important topics in spectral theory and partial differential equations such as inverse problems, both analytical and algebraic; minimal partitions and Pleijel's Theorem; spectral theory for a model in Quantum Field Theory; and beams on Zoll manifolds.

Download or read book Spectral Theory of Infinite Area Hyperbolic Surfaces written by David Borthwick and published by Springer Science & Business Media. This book was released on 2007-09-13 with total page 355 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a self-contained monograph on spectral theory for non-compact Riemann surfaces, focused on the infinite-volume case. By focusing on the scattering theory of hyperbolic surfaces, this work provides a compelling introductory example which will be accessible to a broad audience. The book opens with an introduction to the geometry of hyperbolic surfaces. Then a thorough development of the spectral theory of a geometrically finite hyperbolic surface of infinite volume is given. The final sections include recent developments for which no thorough account exists.

Download or read book Spectral Theory written by David Borthwick and published by Springer Nature. This book was released on 2020-03-12 with total page 339 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Method of Spectral Mappings in the Inverse Problem Theory written by Vacheslav A. Yurko and published by Walter de Gruyter. This book was released on 2013-10-10 with total page 316 pages. Available in PDF, EPUB and Kindle. Book excerpt: Inverse problems of spectral analysis consist in recovering operators from their spectral characteristics. Such problems often appear in mathematics, mechanics, physics, electronics, geophysics, meteorology and other branches of natural science. This monograph is devoted to inverse problems of spectral analysis for ordinary differential equations. Its aim ist to present the main results on inverse spectral problems using the so-called method of spectral mappings, which is one of the main tools in inverse spectral theory. The book consists of three chapters: In Chapter 1 the method of spectral mappings is presented in the simplest version for the Sturm-Liouville operator. In Chapter 2 the inverse problem of recovering higher-order differential operators of the form, on the half-line and on a finite interval, is considered. In Chapter 3 inverse spectral problems for differential operators with nonlinear dependence on the spectral parameter are studied.

Download or read book The Spectral Theory of Geometrically Periodic Hyperbolic 3 Manifolds written by Charles L. Epstein and published by American Mathematical Soc.. This book was released on 1985 with total page 174 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Inverse Spectral Theory written by Jurgen Poschel and published by Academic Press. This book was released on 1987-03-16 with total page 209 pages. Available in PDF, EPUB and Kindle. Book excerpt: Inverse Spectral Theory

Download or read book Spectral Theory and Mathematical Physics A Festschrift in Honor of Barry Simon s 60th Birthday written by Fritz Gesztesy and published by American Mathematical Soc.. This book was released on 2007 with total page 528 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Maxwell Equation Inverse Scattering In Electromagnetism written by Hiroshi Isozaki and published by World Scientific. This book was released on 2018-07-12 with total page 300 pages. Available in PDF, EPUB and Kindle. Book excerpt: How can one determine the physical properties of the medium or the geometrical properties of the domain by observing electromagnetic waves? To answer this fundamental problem in mathematics and physics, this book leads the reader to the frontier of inverse scattering theory for electromagnetism.The first three chapters, written comprehensively, can be used as a textbook for undergraduate students. Beginning with elementary vector calculus, this book provides fundamental results for wave equations and Helmholtz equations, and summarizes the potential theory. It also explains the cohomology theory in an easy and straightforward way, which is an essential part of electromagnetism related to geometry. It then describes the scattering theory for the Maxwell equation by the time-dependent method and also by the stationary method in a concise, but almost self-contained manner. Based on these preliminary results, the book proceeds to the inverse problem for the Maxwell equation.The chapters for the potential theory and elementary cohomology theory are good introduction to graduate students. The results in the last chapter on the inverse scattering for the medium and the determination of Betti numbers are new, and will give a current scope for the inverse spectral problem on non-compact manifolds. It will be useful for young researchers who are interested in this field and trying to find new problems.

Download or read book The Spectral Measure on Non trapping Asymptotically Hyperbolic Manifolds written by Chen, Xi and published by . This book was released on 2015 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis we systematically discuss the spectral measure on non-trapping asymptotically hyperbolic manifolds and related applications in spectral multipliers and Schrodinger equations. Spectral measure is a notion from spectral theory and can be studied via resolvent by Stone's formula. Following the works, due to Mazzeo and Melrose, Melrose, Sa Barreto and Vasy, we construct high energy resolvent by techniques such as the 0-calculus, semiclassical Fourier integral operators, semiclassical intersecting Lagrangian distributions. Borrowing the pseudo differential operator microlocalization tricks, formulated by Guillarmou, Hassell and Sikora, we then prove the spectral measure estimates for large spectral parameters. From the perspective of harmonic analysis, spectral measure is also tied to restriction theorem, which plays a key role in the theory of spectral multipliers. This trilateral relationship is formulated by Guillarmou, Hassell and Sikora. We apply their theory and get restriction theorem for high energy and weak restriction theorem for low energy, together with a crude Lp + L2 boundedness of spectral multipliers, though the spectral measure on asymptotically hyperbolic manifolds is not so ideal as it requires. From dispersive equations' point of view, spectral measure is a cornerstone of Schrodinger propagator. Our spectral measure estimates apply to dispersive estimates for microlocalized high energy truncated propagators for short time as in the work of Hassell and Zhang. Noting the discrepancy of the spectral measure between low and high energy, we also prove the long time dispersive estimates and low energy truncated estimates for short time. By modified Keel-Tao bilinear arguments, due to Anker and Pierfelice, we obtain Strichartz estimates.

Download or read book Spectral Asymptotics on Degenerating Hyperbolic 3 Manifolds written by Józef Dodziuk and published by American Mathematical Soc.. This book was released on 1998 with total page 90 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Spectral Theory of Infinite Area Hyperbolic Surfaces written by David Borthwick and published by Birkhäuser. This book was released on 2016-07-12 with total page 471 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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)

Download or read book Spectral Theory and Differential Equations written by E. Khruslov and published by American Mathematical Society. This book was released on 2014-09-26 with total page 266 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is dedicated to V. A. Marchenko on the occasion of his 90th birthday. It contains refereed original papers and survey articles written by his colleagues and former students of international stature and focuses on the areas to which he made important contributions: spectral theory of differential and difference operators and related topics of mathematical physics, including inverse problems of spectral theory, homogenization theory, and the theory of integrable systems. The papers in the volume provide a comprehensive account of many of the most significant recent developments in that broad spectrum of areas.

Download or read book An Introduction to Spectral Theory written by Andrei Giniatoulline and published by R.T. Edwards, Inc.. This book was released on 2005 with total page 212 pages. Available in PDF, EPUB and Kindle. Book excerpt: A brief and accessible introduction to the spectral theory of linear second order elliptic differential operators. By introducing vital topics of abstract functional analysis where necessary, and using clear and simple proofs, the book develops an elegant presentation of the theory while integrating applications of basic real world problems involving the Laplacian. Suitable for use as a self-contained introduction for beginners or as a one-semester student text; contains some 25 examples and 60 exercises, most with detailed hints.

Download or read book A Spectral Theory for Simply Periodic Solutions of the Sinh Gordon Equation written by Sebastian Klein and published by Springer. This book was released on 2018-12-05 with total page 326 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book develops a spectral theory for the integrable system of 2-dimensional, simply periodic, complex-valued solutions u of the sinh-Gordon equation. Such solutions (if real-valued) correspond to certain constant mean curvature surfaces in Euclidean 3-space. Spectral data for such solutions are defined (following ideas of Hitchin and Bobenko) and the space of spectral data is described by an asymptotic characterization. Using methods of asymptotic estimates, the inverse problem for the spectral data is solved along a line, i.e. the solution u is reconstructed on a line from the spectral data. Finally, a Jacobi variety and Abel map for the spectral curve are constructed and used to describe the change of the spectral data under translation of the solution u. The book's primary audience will be research mathematicians interested in the theory of infinite-dimensional integrable systems, or in the geometry of constant mean curvature surfaces.

Download or read book Inverse Spectral Theory written by Jürgen Pöschel and published by . This book was released on 1987 with total page 192 pages. Available in PDF, EPUB and Kindle. Book excerpt: