The Method Of Rigged Spaces In Singular Perturbation Theory Of Self Adjoint Operators
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| Author | : Volodymyr Koshmanenko |
| Publisher | : Birkhäuser |
| Total Pages | : 251 |
| Release | : 2016-07-08 |
| Genre | : Mathematics |
| ISBN | : 3319295357 |
This monograph presents the newly developed method of rigged Hilbert spaces as a modern approach in singular perturbation theory. A key notion of this approach is the Lax-Berezansky triple of Hilbert spaces embedded one into another, which specifies the well-known Gelfand topological triple. All kinds of singular interactions described by potentials supported on small sets (like the Dirac δ-potentials, fractals, singular measures, high degree super-singular expressions) admit a rigorous treatment only in terms of the equipped spaces and their scales. The main idea of the method is to use singular perturbations to change inner products in the starting rigged space, and the construction of the perturbed operator by the Berezansky canonical isomorphism (which connects the positive and negative spaces from a new rigged triplet). The approach combines three powerful tools of functional analysis based on the Birman-Krein-Vishik theory of self-adjoint extensions of symmetric operators, the theory of singular quadratic forms, and the theory of rigged Hilbert spaces. The book will appeal to researchers in mathematics and mathematical physics studying the scales of densely embedded Hilbert spaces, the singular perturbations phenomenon, and singular interaction problems.
| Author | : Fritz Gesztesy |
| Publisher | : American Mathematical Society |
| Total Pages | : 946 |
| Release | : 2024-09-24 |
| Genre | : Mathematics |
| ISBN | : 1470476665 |
This book provides a detailed treatment of the various facets of modern Sturm?Liouville theory, including such topics as Weyl?Titchmarsh theory, classical, renormalized, and perturbative oscillation theory, boundary data maps, traces and determinants for Sturm?Liouville operators, strongly singular Sturm?Liouville differential operators, generalized boundary values, and Sturm?Liouville operators with distributional coefficients. To illustrate the theory, the book develops an array of examples from Floquet theory to short-range scattering theory, higher-order KdV trace relations, elliptic and algebro-geometric finite gap potentials, reflectionless potentials and the Sodin?Yuditskii class, as well as a detailed collection of singular examples, such as the Bessel, generalized Bessel, and Jacobi operators. A set of appendices contains background on the basics of linear operators and spectral theory in Hilbert spaces, Schatten?von Neumann classes of compact operators, self-adjoint extensions of symmetric operators, including the Friedrichs and Krein?von Neumann extensions, boundary triplets for ODEs, Krein-type resolvent formulas, sesquilinear forms, Nevanlinna?Herglotz functions, and Bessel functions.
| Author | : Raul E Curto |
| Publisher | : Springer Nature |
| Total Pages | : 531 |
| Release | : 2020-12-12 |
| Genre | : Mathematics |
| ISBN | : 3030433803 |
This book is the proceeding of the International Workshop on Operator Theory and Applications (IWOTA) held in July 2018 in Shanghai, China. It consists of original papers, surveys and expository articles in the broad areas of operator theory, operator algebras and noncommutative topology. Its goal is to give graduate students and researchers a relatively comprehensive overview of the current status of research in the relevant fields. The book is also a special volume dedicated to the memory of Ronald G. Douglas who passed away on February 27, 2018 at the age of 79. Many of the contributors are Douglas’ students and past collaborators. Their articles attest and commemorate his life-long contribution and influence to these fields.
| Author | : Yurij M. Berezansky |
| Publisher | : Springer Nature |
| Total Pages | : 489 |
| Release | : 2024-01-06 |
| Genre | : Mathematics |
| ISBN | : 3031463870 |
This monograph presents the solution of the classical moment problem, the construction of Jacobi matrices and corresponding polynomials. The cases of strongly,trigonometric, complex and real two-dimensional moment problems are discussed, and the Jacobi-type matrices corresponding to the trigonometric moment problem are shown. The Berezansky theory of the expansion in generalized eigenvectors for corresponding set of commuting operators plays the key role in the proof of results. The book is recommended for researchers in fields of functional analysis, operator theory, mathematical physics, and engineers who deal with problems of coupled pendulums.
| Author | : Volodymyr Koshmanenko |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 316 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 9401146195 |
The notion of singular quadratic form appears in mathematical physics as a tool for the investigation of formal expressions corresponding to perturbations devoid of operator sense. Numerous physical models are based on the use of Hamiltonians containing perturba tion terms with singular properties. Typical examples of such expressions are Schrodin ger operators with O-potentials (-~ + AD) and Hamiltonians in quantum field theory with perturbations given in terms of operators of creation and annihilation (P(
| Author | : |
| Publisher | : |
| Total Pages | : 546 |
| Release | : 2007 |
| Genre | : Mathematical physics |
| ISBN | : |
| Author | : |
| Publisher | : |
| Total Pages | : 410 |
| Release | : 2006 |
| Genre | : Functional analysis |
| ISBN | : |
| Author | : Michael Reed |
| Publisher | : Elsevier |
| Total Pages | : 344 |
| Release | : 2012-12-02 |
| Genre | : Science |
| ISBN | : 0323155006 |
Methods of Modern Mathematical Physics, Volume I: Functional Analysis discusses the fundamental principles of functional analysis in modern mathematical physics. This book also analyzes the influence of mathematics on physics, such as the Newtonian mechanics used to interpret all physical phenomena. Organized into eight chapters, this volume starts with an overview of the functional analysis in the study of several concrete models. This book then discusses how to generalize the Lebesgue integral to work with functions on the real line and with Borel sets. This text also explores the properties of finite-dimensional vector spaces. Other chapters discuss the normed linear spaces, which have the property of being complete. This monograph further examines the general class of topologized vector spaces and the spaces of distributions that arise in a wide variety of physical problems and functional situations. This book is a valuable resource for mathematicians and physicists. Students and researchers in the field of geometry will also find this book extremely useful.
| Author | : Sergio Albeverio |
| Publisher | : Birkhäuser |
| Total Pages | : 423 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 303488219X |
CO«i»b.H BaCHJIbeBHa lU>BaJIeBcR8JI (Sonja Kovalevsky) was born in Moscow in 1850 and died in Stockholm in 1891. Between these years, in the then changing and turbulent circumstances for Europe, lies the all too brief life of this remarkable woman. This life was lived out within the great European centers of power and learning in Russia, France, Germany, Switzerland, England and Sweden. To this day, now 150 years after her birth, her influence for and contribution to mathe matics, science, literature, women's rights and democratic government are recorded and reviewed, not only in Europe but now in countries far removed in time and distance from the lands of her birth and being. This volume, dedicated to her memory and to her achievements, records the Proceedings of the Marcus Wallenberg Symposium held, in memory of Sonja Kovalevsky, at Stockholm University from 18 to 22 June 2000. The symposium was held at the Department of Mathematics with its excellent library and lecture halls providing favourable working conditions. Within these pages are contained a curriculum vitae for Sonja Kovalevsky, a list of all her scientific publications, together with a copy of the moving and elegant obituary notice written by her friend and protector Gosta Mittag-Leffler. These papers are followed by a leading article entitled Sonja Kovalevsky: Her life and professorship in Stockholm, written especially for this volume by Jan-Erik Bjork in preparation for his major address to the Symposium.
| Author | : Vadim Adamyan |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 497 |
| Release | : 2009-08-29 |
| Genre | : Mathematics |
| ISBN | : 3764399198 |
This is the first of two volumes containing peer-reviewed research and survey papers based on talks at the International Conference on Modern Analysis and Applications. The papers describe the contemporary development of subjects influenced by Mark Krein.
