Overview
- Deals with singular quadratic forms and singular perturbations of self-adjoint operators considered in rigged Hilbert spaces
- Demonstrates the theory of singular perturbations by three notions into a unique object with a three-face image existing in the rigged Hilbert space
- Provides an adequate tool for exploration of the singular perturbation problem
Part of the book series: Operator Theory: Advances and Applications (OT, volume 253)
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Table of contents (9 chapters)
Keywords
About this book
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.
Reviews
“This well written book is a very welcome addition, filling a significant gap in the literature on singular perturbation theory.” (Jaydeb Sarkar, zbMATH 1447.47010, 2020)
Authors and Affiliations
About the authors
Bibliographic Information
Book Title: The Method of Rigged Spaces in Singular Perturbation Theory of Self-Adjoint Operators
Authors: Volodymyr Koshmanenko, Mykola Dudkin
Translated by: Nataliia Koshmanenko
Series Title: Operator Theory: Advances and Applications
DOI: https://doi.org/10.1007/978-3-319-29535-0
Publisher: Birkhäuser Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing Switzerland 2016
Hardcover ISBN: 978-3-319-29533-6Published: 19 July 2016
Softcover ISBN: 978-3-319-80592-4Published: 31 May 2018
eBook ISBN: 978-3-319-29535-0Published: 08 July 2016
Series ISSN: 0255-0156
Series E-ISSN: 2296-4878
Edition Number: 1
Number of Pages: XX, 237
Number of Illustrations: 1 b/w illustrations
Additional Information: Original Ukrainian edition published by Institute of Mathematics, NAS of Ukraine, Kyiv, 2013
Topics: Operator Theory, Measure and Integration, Mathematical Applications in the Physical Sciences