The Method of Rigged Spaces in Singular Perturbation Theory of Self-Adjoint Operators

  • Volodymyr Koshmanenko
  • Mykola Dudkin

Part of the Operator Theory: Advances and Applications book series (OT, volume 253)

Table of contents

  1. Front Matter
    Pages i-xx
  2. Volodymyr Koshmanenko, Mykola Dudkin
    Pages 1-15
  3. Volodymyr Koshmanenko, Mykola Dudkin
    Pages 17-36
  4. Volodymyr Koshmanenko, Mykola Dudkin
    Pages 37-59
  5. Volodymyr Koshmanenko, Mykola Dudkin
    Pages 61-71
  6. Volodymyr Koshmanenko, Mykola Dudkin
    Pages 73-90
  7. Volodymyr Koshmanenko, Mykola Dudkin
    Pages 91-111
  8. Volodymyr Koshmanenko, Mykola Dudkin
    Pages 113-167
  9. Volodymyr Koshmanenko, Mykola Dudkin
    Pages 169-191
  10. Volodymyr Koshmanenko, Mykola Dudkin
    Pages 193-219
  11. Back Matter
    Pages 221-237

About this book

Introduction

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.

Keywords

densely embedded Hilbert space rigged Hilbert space self-adjoint operator self-adjoint extension singular quadratic form singularly perturbed operator

Authors and affiliations

  • Volodymyr Koshmanenko
    • 1
  • Mykola Dudkin
    • 2
  1. 1.National Academy of Sciences of Ukraine Institute of MathematicsKyivUkraine
  2. 2.Kyiv Polytechnic InstituteNational Technical University of UkraineKyivUkraine

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-29535-0
  • Copyright Information Springer International Publishing Switzerland 2016
  • Publisher Name Birkhäuser, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-29533-6
  • Online ISBN 978-3-319-29535-0
  • Series Print ISSN 0255-0156
  • Series Online ISSN 2296-4878
  • About this book