Download or read book Inverse Spectral Problems for Linear Differential Operators and Their Applications written by V A Yurko and published by CRC Press. This book was released on 2000-01-18 with total page 272 pages. Available in PDF, EPUB and Kindle. Book excerpt: Aims to construct the inverse problem theory for ordinary non-self-adjoint differential operators of arbitary order on the half-line and on a finite interval. The book consists of two parts: in the first part the author presents a general inverse problem of recovering differential equations with integrable coefficients when the behaviour of the spe
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 Method of Spectral Mappings in the Inverse Problem Theory written by V. A. Yurko and published by . This book was released on 2002 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 Inverse Sturm Liouville Problems and Their Applications written by G. Freiling and published by Nova Biomedical Books. This book was released on 2001 with total page 324 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents the main results and methods on inverse spectral problems for Sturm-Liouville differential operators and their applications. 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 sciences. Inverse problems also play an important role in solving non-linear evolution equations in mathematical physics. Interest in this subject has been increasing permanently because of the appearance of new important applications, resulting in intensive study of inverse problem theory all over the world.
Download or read book Spectral Analysis of Differential Operators written by Fedor S. Rofe-Beketov and published by World Scientific. This book was released on 2005 with total page 466 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first monograph devoted to the Sturm oscillatory theory for infinite systems of differential equations and its relations with the spectral theory. It aims to study a theory of self-adjoint problems for such systems, based on an elegant method of binary relations. Another topic investigated in the book is the behavior of discrete eigenvalues which appear in spectral gaps of the Hill operator and almost periodic SchrAdinger operators due to local perturbations of the potential (e.g., modeling impurities in crystals). The book is based on results that have not been presented in other monographs. The only prerequisites needed to read it are basics of ordinary differential equations and operator theory. It should be accessible to graduate students, though its main topics are of interest to research mathematicians working in functional analysis, differential equations and mathematical physics, as well as to physicists interested in spectral theory of differential operators."
Download or read book Inverse Problems and Spectral Theory written by Hiroshi Isozaki and published by American Mathematical Soc.. This book was released on 2004 with total page 258 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume grew out of a workshop on spectral theory of differential operators and inverse problems held at the Research Institute for Mathematical Sciences (Kyoto University). The gathering of nearly 100 participants at the conference suggests the increasing interest in this field of research. The focus of the book is on spectral theory for differential operators and related inverse problems. It includes selected topics from the following areas: electromagnetism, elasticity, the Schrodinger equation, differential geometry, and numerical analysis. The material is suitable for graduate students and researchers interested in inverse problems and their applications.
Download or read book Spectral Analysis of Differential Operators written by Fedor S Rofe-Beketov and published by World Scientific. This book was released on 2005-08-29 with total page 464 pages. Available in PDF, EPUB and Kindle. Book excerpt: ' This is the first monograph devoted to the Sturm oscillatory theory for infinite systems of differential equations and its relations with the spectral theory. It aims to study a theory of self-adjoint problems for such systems, based on an elegant method of binary relations. Another topic investigated in the book is the behavior of discrete eigenvalues which appear in spectral gaps of the Hill operator and almost periodic Schrödinger operators due to local perturbations of the potential (e.g., modeling impurities in crystals). The book is based on results that have not been presented in other monographs. The only prerequisites needed to read it are basics of ordinary differential equations and operator theory. It should be accessible to graduate students, though its main topics are of interest to research mathematicians working in functional analysis, differential equations and mathematical physics, as well as to physicists interested in spectral theory of differential operators. Contents:Relation Between Spectral and Oscillatory Properties for the Matrix Sturm–Liouville ProblemFundamental System of Solutions for an Operator Differential Equation with a Singular Boundary ConditionDependence of the Spectrum of Operator Boundary Problems on Variations of a Finite or Semi-Infinite IntervalRelation Between Spectral and Oscillatory Properties for Operator Differential Equations of Arbitrary OrderSelf-Adjoint Extensions of Systems of Differential Equations of Arbitrary Order on an Infinite Interval in the Absolutely Indefinite CaseDiscrete Levels in Spectral Gaps of Perturbed Schrödinger and Hill Operators Readership: Graduate students, mathematicians and physicists interested in functional analysis, differential equations and mathematical physics. Keywords:Operator;Differential Equation;Self-Adjoint Extension;Spectrum;Perturbation;OscillationKey Features:Detailed bibliographical comments and some open questions are given after each chapterIndicates connections between the content of the book and many other topics in mathematics and physicsOpen questions are formulated and commented with the intention to attract attention of young mathematiciansReviews:“The appendix is very valuable and helps the reader to find an orientation in the very voluminous literature devoted to the spectral theory of differential operators … anybody interested in the spectral theory of differential operators will find interesting information in the book, including formulation of open problems for possible investigation.”Mathematical Reviews “This book is well-written, and a list of symbols and the index prove useful. A substantial number of open questions is also included. Although addressed primarily to the research community, the book could also be used as a graduate textbooks.”Zentralblatt MATH '
Download or read book Spectral Theory and Differential Operators written by David Edmunds and published by Oxford University Press. This book was released on 2018-05-03 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is an updated version of the classic 1987 monograph "Spectral Theory and Differential Operators".The original book was a cutting edge account of the theory of bounded and closed linear operators in Banach and Hilbert spaces relevant to spectral problems involving differential equations. It is accessible to a graduate student as well as meeting the needs of seasoned researchers in mathematics and mathematical physics. This revised edition corrects various errors, and adds extensive notes to the end of each chapter which describe the considerable progress that has been made on the topic in the last 30 years.
Download or read book Inverse Sturm Liouville Problems written by B. M. Levitan and published by Walter de Gruyter GmbH & Co KG. This book was released on 2018-07-12 with total page 252 pages. Available in PDF, EPUB and Kindle. Book excerpt: The interest in inverse problems of spectral analysis has increased considerably in recent years due to the applications to important non-linear equations in mathematical physics. This monograph is devoted to the detailed theory of inverse problems and methods of their solution for the Sturm-Liouville case. Chapters 1--6 contain proofs which are, in many cases, very different from those known earlier. Chapters 4--6 are devoted to inverse problems of quantum scattering theory with attention being focused on physical applications. Chapters 7--11 are based on the author's recent research on the theory of finite- and infinite-zone potentials. A chapter discussing the applications to the Korteweg--de Vries problem is also included. This monograph is important reading for all researchers in the field of mathematics and physics.
Download or read book Spectral Geometry of Partial Differential Operators written by Michael Ruzhansky and published by CRC Press. This book was released on 2020-02-07 with total page 366 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of Spectral Geometry of Partial Differential Operators is to provide a basic and self-contained introduction to the ideas underpinning spectral geometric inequalities arising in the theory of partial differential equations. Historically, one of the first inequalities of the spectral geometry was the minimization problem of the first eigenvalue of the Dirichlet Laplacian. Nowadays, this type of inequalities of spectral geometry have expanded to many other cases with number of applications in physics and other sciences. The main reason why the results are useful, beyond the intrinsic interest of geometric extremum problems, is that they produce a priori bounds for spectral invariants of (partial differential) operators on arbitrary domains. Features: Collects the ideas underpinning the inequalities of the spectral geometry, in both self-adjoint and non-self-adjoint operator theory, in a way accessible by anyone with a basic level of understanding of linear differential operators Aimed at theoretical as well as applied mathematicians, from a wide range of scientific fields, including acoustics, astronomy, MEMS, and other physical sciences Provides a step-by-step guide to the techniques of non-self-adjoint partial differential operators, and for the applications of such methods. Provides a self-contained coverage of the traditional and modern theories of linear partial differential operators, and does not require a previous background in operator theory.
Download or read book Spectral Theory and Differential Operators written by David Eric Edmunds and published by . This book was released on 1987 with total page 604 pages. Available in PDF, EPUB and Kindle. Book excerpt: This comprehensive and long-awaited volume provides an up-to-date account of those parts of the theory of bounded and closed linear operators in Banach and Hilbert spaces relevant to spectral problems involving differential equations. For the first time it brings together recent results in essential spectra, measures of non-compactness, entropy numbers, approximation numbers, eigenvalues, and the relationships among these concepts. The authors illustrate abstract theory with results for embedding maps between Sobolev spaces. Strong emphasis is placed on application to boundary-value problems for general second-order linear elliptic equations in an arbitrary domain in Rn. The book introduces some key eastern European work, never before available in English translation.
Download or read book Differential Operators and Related Topics written by V. M. Adami͡an and published by Springer Science & Business Media. This book was released on 2000 with total page 438 pages. Available in PDF, EPUB and Kindle. Book excerpt: About the Mark Krein International Conference.- Mark Grigorevich Krein (A short biography).- The Seminar on Ship Hydrodynamics, Organized by M.G. Krein.- Review Papers: The Works of M.G. Krein on Eigenfunction Expansion for Selfadjoint Operators and their Applications and Development.- M.G. Krein and the Extension Theory of Symmetric Operators. Theory of Entire Operators.- Works by M.G. Krein on Inverse Problems.- Research Papers: The Spectrum of Periodic Point Perturbations and the Krein Resolvent Formula.- The Periodic Choquard Equation.- On the Best Constant in a Poincare-Sobolev Inequality.- On Solutions of Parabolic Equations from Families of Banach Spaces Dependent on Time.- Canonical Systems on the Line with Rational Spectral Densities: Explicit Formulas.- Oscillations in Systems with Periodic Coefficients and Sector-restricted Nonlinearities.- Differential Operator Matrices of Mixed Order with Periodic Coefficients.- Asymptotics of Generalized Eigenvectors for Unbounded Jacobi Matrices with Power-like Weights, Pauli Matrices Commutation Relations and Cesaro Averaging.- Functional Means, Convolution Operators and Semigroups.- The Inverse Spectral Problem for First Order Systems on the Half Line.- Exact Solution of the Marchenko Equation Relevant to Inverse Scattering on the Line.- An Arbitrary Oriented Crack in the Box Shell.- Homogeneity of a String having Three Unperturbed Spectra.- On the Integro-differential Equation of a Torsion of an Elastic Medium Including a Cylindrical Crack.- Green's Formula and Theorems on Isomorphisms for General Elliptic Problems for Douglis-Nirenberg Elliptic Systems.- Sobolev's Problem in Complete Scale of Banach Spaces.- On the Simple Waves with Profiles in the Form of some Special Functions-Chebyshov-Hermite, Mathiev, Whittaker-in Two-phase Media.- Inverse Spectral Problem Related to the N-wave Equation.- Degenerated Hyperbolic Approximations of the Wave Theory of Elastic Plates.- Elliptic Problems with a Shift in Complete Scales of Sobolev-type Spaces.- On the Extremal Regularization of the Variational Inequalities with Multivalued Operators.- Poly-Fock Spaces.- Diffraction of Longitudinal Shear Waves by a Hollow Thick Circular Cylinder which is Situated in the Elastic Halfspace.- On M.G. Krein's Spectral Shift Function for Canonical Systems of Differential Equations.- Table of Contents of Volume II.
Download or read book Spectral Theory of Ordinary Differential Operators written by Joachim Weidmann and published by Lecture Notes in Mathematics. This book was released on 1987-05-06 with total page 316 pages. Available in PDF, EPUB and Kindle. Book excerpt: These notes will be useful and of interest to mathematicians and physicists active in research as well as for students with some knowledge of the abstract theory of operators in Hilbert spaces. They give a complete spectral theory for ordinary differential expressions of arbitrary order n operating on -valued functions existence and construction of self-adjoint realizations via boundary conditions, determination and study of general properties of the resolvent, spectral representation and spectral resolution. Special attention is paid to the question of separated boundary conditions, spectral multiplicity and absolutely continuous spectrum. For the case nm=2 (Sturm-Liouville operators and Dirac systems) the classical theory of Weyl-Titchmarch is included. Oscillation theory for Sturm-Liouville operators and Dirac systems is developed and applied to the study of the essential and absolutely continuous spectrum. The results are illustrated by the explicit solution of a number of particular problems including the spectral theory one partical Schrödinger and Dirac operators with spherically symmetric potentials. The methods of proof are functionally analytic wherever possible.
Download or read book An Introduction to Inverse Scattering and Inverse Spectral Problems written by Khosrow Chadan and published by SIAM. This book was released on 1997-01-01 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt: Here is a clearly written introduction to three central areas of inverse problems: inverse problems in electromagnetic scattering theory, inverse spectral theory, and inverse problems in quantum scattering theory. Inverse problems, one of the most attractive parts of applied mathematics, attempt to obtain information about structures by nondestructive measurements. Based on a series of lectures presented by three of the authors, all experts in the field, the book provides a quick and easy way for readers to become familiar with the area through a survey of recent developments in inverse spectral and inverse scattering problems.
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 Convolution like Structures Differential Operators and Diffusion Processes written by Rúben Sousa and published by Springer Nature. This book was released on 2022-07-27 with total page 269 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an introduction to recent developments in the theory of generalized harmonic analysis and its applications. It is well known that convolutions, differential operators and diffusion processes are interconnected: the ordinary convolution commutes with the Laplacian, and the law of Brownian motion has a convolution semigroup property with respect to the ordinary convolution. Seeking to generalize this useful connection, and also motivated by its probabilistic applications, the book focuses on the following question: given a diffusion process Xt on a metric space E, can we construct a convolution-like operator * on the space of probability measures on E with respect to which the law of Xt has the *-convolution semigroup property? A detailed analysis highlights the connection between the construction of convolution-like structures and disciplines such as stochastic processes, ordinary and partial differential equations, spectral theory, special functions and integral transforms. The book will be valuable for graduate students and researchers interested in the intersections between harmonic analysis, probability theory and differential equations.
Download or read book Transmutations Singular and Fractional Differential Equations with Applications to Mathematical Physics written by Elina Shishkina and published by Academic Press. This book was released on 2020-07-24 with total page 594 pages. Available in PDF, EPUB and Kindle. Book excerpt: Transmutations, Singular and Fractional Differential Equations with Applications to Mathematical Physics connects difficult problems with similar more simple ones. The book's strategy works for differential and integral equations and systems and for many theoretical and applied problems in mathematics, mathematical physics, probability and statistics, applied computer science and numerical methods. In addition to being exposed to recent advances, readers learn to use transmutation methods not only as practical tools, but also as vehicles that deliver theoretical insights. Presents the universal transmutation method as the most powerful for solving many problems in mathematics, mathematical physics, probability and statistics, applied computer science and numerical methods Combines mathematical rigor with an illuminating exposition full of historical notes and fascinating details Enables researchers, lecturers and students to find material under the single "roof"
