Overview

This monograph is devoted to the systematic presentation of the method of singular quadratic forms in the perturbation theory of self-adjoint operators. The concept of a singular (nowhere closable) quadratic form, a key notion of the present volume, is treated from different points of view such as definition, properties, relations with regular (closable) quadratic forms, operator representation, classification in the scale of Hilbert spaces and especially as an object carrying a singular perturbation for Hamiltonians. The main idea is to interpret singular quadratic form in the role of an abstract boundary condition for self-adjoint extension. Various aspects of the singularity principle are investigated, such as the construction of singularly perturbed operators, higher powers of perturbed operators, the transition to a new orthogonally extended state space, as well as approximation and regularization. Furthermore, applications dealing with singular Wick monomials in the Fock space and mathematical scattering theory are included.

Full Product Details

Author:   Volodymyr Koshmanenko
Publisher:   Springer
Imprint:   Springer
Edition:   1999 ed.
Volume:   474
Dimensions:   Width: 17.00cm , Height: 1.90cm , Length: 24.40cm
Weight:   1.390kg
ISBN:  

9780792356257

ISBN 10:   079235625
Pages:   312
Publication Date:   30 April 1999
Audience:   College/higher education ,  Professional and scholarly ,  Undergraduate ,  Postgraduate, Research & Scholarly
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

1. Quadratic Forms and Linear Operators.- 1. Preliminary Facts about Quadratic Forms.- 2. Closed and Closable Quadratic Forms.- 3. Operator Representations of Quadratic Forms.- 4. Quadratic Forms in the Theory of Self-Adjoint Extensions of Symmetric Operators.- 2. Singular Quadratic Forms.- 5. Definition of Singular Quadratic Forms.- 6. Properties of Singular Quadratic Forms.- 7. Operator Representation of Singular Quadratic Forms.- 8. Singular Quadratic Forms in the A-Scale of Hilbert Spaces.- 9. Regularization.- 3. Singular Perturbations of Self-Adjoint Operators.- 10. Rank-One Singular Perturbations.- 11. Singular Perturbations of Finite Rank.- 12. Method of Self-Adjoint Extensions.- 13. Powers of Singularly Perturbed Operators.- 14. Method of Orthogonal Extensions.- 15. Approximations.- 4. Applications to Quantum Field Theory.- 16. Singular Properties of Wick Monomials.- 17. Orthogonally Extended Fock Space.- 18. Scattering and Spectral Problems.- References.- Notation.

Reviews

' ...this text is a valuable contribution to the literature on the abstract aspects of singular perturbation theory.' Mathematical Reviews, 2001a

' ...this text is a valuable contribution to the literature on the abstract aspects of singular perturbation theory.' Mathematical Reviews, 2001a

...this text is a valuable contribution to the literature on the abstract aspects of singular perturbation theory.' Mathematical Reviews, 2001a

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