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| Author | : Vladimir Aleksandrovich Marchenko |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 410 |
| Release | : 2011-04-27 |
| Genre | : Mathematics |
| ISBN | : 0821853163 |
The spectral theory of Sturm-Liouville operators is a classical domain of analysis, comprising a wide variety of problems. This book aims to show what can be achieved with the aid of transformation operators in spectral theory as well as their applications.
| Author | : V.A. Marchenko |
| Publisher | : Springer-Verlag |
| Total Pages | : 379 |
| Release | : 2013-11-21 |
| Genre | : Science |
| ISBN | : 3034854854 |
| Author | : Werner O. Amrein |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 348 |
| Release | : 2005-12-05 |
| Genre | : Mathematics |
| ISBN | : 3764373598 |
This is a collection of survey articles based on lectures presented at a colloquium and workshop in Geneva in 2003 to commemorate the 200th anniversary of the birth of Charles François Sturm. It aims at giving an overview of the development of Sturm-Liouville theory from its historical roots to present day research. It is the first time that such a comprehensive survey has been made available in compact form. The contributions come from internationally renowned experts and cover a wide range of developments of the theory. The book can therefore serve both as an introduction to Sturm-Liouville theory and as background for ongoing research. The volume is addressed to researchers in related areas, to advanced students and to those interested in the historical development of mathematics. The book will also be of interest to those involved in applications of the theory to diverse areas such as engineering, fluid dynamics and computational spectral analysis.
| Author | : G. Freiling |
| Publisher | : Nova Biomedical Books |
| Total Pages | : 324 |
| Release | : 2001 |
| Genre | : Mathematics |
| ISBN | : |
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.
| Author | : Vladimir Marchenko |
| Publisher | : Birkhäuser |
| Total Pages | : 367 |
| Release | : 2014-08-23 |
| Genre | : Science |
| ISBN | : 9783034854863 |
The development of many important directions of mathematics and physics owes a major debt to the concepts and methods which evolved during the investigation of such simple objects as the Sturm-Liouville equation 2 2 y" + q(x)y = zy and the allied Sturm-Liouville operator L = - d /dx + q(x) (lately Land q(x) are often termed the one-dimensional Schrödinger operator and the potential). These provided a constant source of new ideas and problems in the spectral theory of operators and kindred areas of analysis. This sourse goes back to the first studies of D. Bernoulli and L. Euler on the solution of the equation describing the vibrations of astring, and still remains productive after more than two hundred years. This is confirmed by the recent discovery, made by C. Gardner, J. Green, M. Kruskal, and R. Miura [6J, of an unexpected connection between the spectral theory of Sturm-Liouville operators and certain nonlinear partial differential evolution equations. The methods used (and often invented) during the study of the Sturm-Liouville equation have been constantly enriched. In the 40's a new investigation tool joined the arsenal - that of transformation operators.
| Author | : Anton Zettl |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 346 |
| Release | : 2005 |
| Genre | : Education |
| ISBN | : 0821852671 |
In 1836-1837 Sturm and Liouville published a series of papers on second order linear ordinary differential operators, which started the subject now known as the Sturm-Liouville problem. In 1910 Hermann Weyl published an article which started the study of singular Sturm-Liouville problems. Since then, the Sturm-Liouville theory remains an intensely active field of research, with many applications in mathematics and mathematical physics. The purpose of the present book is (a) to provide a modern survey of some of the basic properties of Sturm-Liouville theory and (b) to bring the reader to the forefront of knowledge about some aspects of this theory. To use the book, only a basic knowledge of advanced calculus and a rudimentary knowledge of Lebesgue integration and operator theory are assumed. An extensive list of references and examples is provided and numerous open problems are given. The list of examples includes those classical equations and functions associated with the names of Bessel, Fourier, Heun, Ince, Jacobi, Jorgens, Latzko, Legendre, Littlewood-McLeod, Mathieu, Meissner, Morse, as well as examples associated with the harmonic oscillator and the hydrogen atom. Many special functions of applied mathematics and mathematical physics occur in these examples.
| Author | : Boris Moiseevič Levitan |
| Publisher | : VSP |
| Total Pages | : 258 |
| Release | : 1987 |
| Genre | : Mathematics |
| ISBN | : 9789067640558 |
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.
| Author | : Mohammed Al-Gwaiz |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 270 |
| Release | : 2008-01-15 |
| Genre | : Mathematics |
| ISBN | : 1846289718 |
Developed from a course taught to senior undergraduates, this book provides a unified introduction to Fourier analysis and special functions based on the Sturm-Liouville theory in L2. The text’s presentation follows a clear, rigorous mathematical style that is highly readable. The author first establishes the basic results of Sturm-Liouville theory and then provides examples and applications to illustrate the theory. The final two chapters, on Fourier and Laplace transformations, demonstrate the use of the Fourier series method for representing functions to integral representations.
| Author | : Fritz Gesztesy |
| Publisher | : |
| Total Pages | : 0 |
| Release | : 2024 |
| Genre | : Operator theory |
| ISBN | : 9781470476663 |
| Author | : Jussi Behrndt |
| Publisher | : Springer Nature |
| Total Pages | : 772 |
| Release | : 2020-01-03 |
| Genre | : Mathematics |
| ISBN | : 3030367142 |
This open access book presents a comprehensive survey of modern operator techniques for boundary value problems and spectral theory, employing abstract boundary mappings and Weyl functions. It includes self-contained treatments of the extension theory of symmetric operators and relations, spectral characterizations of selfadjoint operators in terms of the analytic properties of Weyl functions, form methods for semibounded operators, and functional analytic models for reproducing kernel Hilbert spaces. Further, it illustrates these abstract methods for various applications, including Sturm-Liouville operators, canonical systems of differential equations, and multidimensional Schrödinger operators, where the abstract Weyl function appears as either the classical Titchmarsh-Weyl coefficient or the Dirichlet-to-Neumann map. The book is a valuable reference text for researchers in the areas of differential equations, functional analysis, mathematical physics, and system theory. Moreover, thanks to its detailed exposition of the theory, it is also accessible and useful for advanced students and researchers in other branches of natural sciences and engineering.
