Table Of ContentSingular Quadratic Fonns in Perturbation Theory
Mathematics and Its Applications
Managing Editor:
M.HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 474
Singular Quadratic Forms
in Perturbation Theory
by
Volodymyr Koshmanenko
Institute of Mathematics,
National Academy of Sciences,
Kiev, Ukraine
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-010-5952-7 ISBN 978-94-011-4619-7 (eBook)
DOI 10.1007/978-94-011-4619-7
Printed on acid-free paper
This is a completely revised and updated translation
of the original Russian work Singular Bilinear Forms
in Perturbation Theory of Self-Adjoint Operators,
Kiev, Naukova Dumka, ©1993.
Translated by P.V. Malyshev and D.V. Malyshev.
All Rights Reserved
© 1999 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1999
Softcover reprint of the hardcover 1s t edition 1999
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner
conTEnTS
Preface to the English Edition vii
Introduction 1
Chapter 1. QUADRATIC FORMS AND LINEAR OPERATORS 5
1. Preliminary Facts about Quadratic Forms 5
2. Closed and Closable Quadratic Forms 16
3. Operator Representations of Quadratic Forms 31
4. Quadratic Forms in the Theory of Self-Adjoint Extensions
of Symmetric Operators 47
Chapter 2. SINGULAR QUADRATIC FORMS 59
5. Definition of Singular Quadratic Forms 60
6. Properties of Singular Quadratic Forms 74
7. Operator Representation of Singular Quadratic Forms 81
8. Singular Quadratic Forms in the A-Scale of Hilbert Spaces 87
9. Regularization 102
Chapter 3. SINGULAR PERTURBATIONS
OF SELF -ADJOINT OPERATORS 123
lO. Rank-One Singular Perturbations 124
11. Singular Perturbations of Finite Rank 143
12. Method of Self-Adjoint Extensions 157
13. Powers of Singularly Perturbed Operators 172
14. Method of Orthogonal Extensions 191
15. Approximations 207
v
vi Contents
Chapter 4. APPLICATIONS TO QUANTUM FIELD THEORY 227
16. Singular Properties of Wick Monomials 227
17. Orthogonally Extended Fock Space 238
18. Scattering and Spectral Problems 254
REFERENCES 281
SUBJECT INDEX 305
NOTATION 307
PREFACE TO THE EnGLISH EDITIon
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( <p )-mod
els ).
Traditional approaches to the solution of problems of this kind are based on the idea
of approximation of a singular expression corresponding to a give:n perturbation by a
sequence of regularized perturbations. This approach is inevitably connected with vari
ous ambiguities and divergences. In a certain sense, these problems are solved by renor
malization theory, which, however, is too complicated. This explains the necessity in the
development of other methods in the the theory of singular perturbations. In particular,
the method of singular quadratic forms combined with the method of self-adjoint exten
sions seems to be one of the most promising approaches.
This monograph is, in fact, the first attempt of systematic presentation of the method
of singular quadratic forms in the theory of perturbations of self-adjoint operators.
A major part of the monograph is devoted to the detailed investigation of properties
of singular quadratic forms, which are the key objects of the theory. We consider numer
ous examples of such forms. We give a detailed presentation of several methods for the
construction of a singularly perturbed operator, depending on the degree of singularity of
a perturbation.
As an application, we consider a model of a Hamiltonian of free quantum field in the
Fock space with a perturbation given by singular Wick monomials.
The author assumes that the reader is familiar with such branches of functional ana
lysis and mathematical physics as the theory of linear operators in Hilbert spaces [AkG],
the theory of perturbations of linear operators [Katl], the theory of quadratic forms
[Katl], [ReS I], [ReS2], scattering theory [Katl], [ReS3], the theory of rigged Hilbert
vii
viii Preface to the English Edition
spaces [Berl], [Ber2], the theory of self-adjoint extensions of Hermitian operators
[AkG], [Kre2], and quantum field theory [BLT] and with the remarkable monograph
[AGHKH], where numerous explicitly solvable models with singular perturbations are
considered.
The author expresses his deep gratitude to Prof. S. Albeverio, Prof. Yu. M. Berezan
skii, Prof. L. P. Nizhnik, Prof. Yu. G. Kondratiev, Prof. W. Karwowski, Prof.H. Neid
hardt, Prof. 1. F. Brasche, and Prof. S. 6ta for fruitful discussions helpful remarks, and
collaboration.
InTRODUCTIon
The theory of perturbations of linear operators in Hilbert spaces is, in fact, a theory of
additive perturbations in which a perturbed operator H is defined as the sum H = Ho +
£ V of an operator Ho corresponding to the Hamiltonian of a free physical system and an
operator V that describes interaction; £ is a coupling constant (see the fundamental
monograph of Kato [Katl]). Assume that Ho and V are self-adjoint operators in a Hil
bert space J{ which is the space of states of a physical system. The following problems
are usually regarded as basic problems of perturbation theory: criteria of essential self
adjointness of the operator H, the structure of the spectrum of H and its dependence on
£ and the properties of V, the construction of the resolvent R z :: (H - z )-1 and the
unitary group e-itH which describes the evolution of the physical system in time, and the
existence of the wave operators W± (H, H 0) and the scattering matrix.
The main difference between the theory of singular perturbations and the ordinary
perturbation theory of linear operators is that singular perturbations cannot be regarded as
additive. It is impossible to represent the perturbed operator in the form of a sum of the
operator Ho and the operator that describes interaction. In fact, singular perturbations
correspond to changes in boundary conditions (in an abstract sense). Singular perturba
tions are usually concentrated on subsets of Lebesgue measure zero and, therefore, are
equal to zero in dense domains in J{. Consequently, they do not possess the ordinary
operator representation. At the same time, they are well described in terms of (nonclos
able and singular) quadratic forms.
The notion of singular quadratic forms is the key object of the present monograph.
Roughly speaking, singularity means that the quadratic form is not closed anywhere in
J{ (see Definition 2.l.2). In practice, this property takes place if the form is equal to
zero in a dense domain in the space J{. In general, any densely defined Hermitian quad
ratic form 'Y bounded from below admits the canonical expansion Y= 'Yr + 'Ys into regu
lar (= closed) and singular (= everywhere nonclosed) components (see [Sim4] and
[Kos3]). Here, we mainly consider the case of pure singular perturbations, i.e., 'Y = 'Ys.
A typical example of a problem with singular perturbations is given by the following
model: the Schrodinger operator - ~ + £C) with C)-potential. Very many publications are
devoted to the investigation of this model and its various modifications and versions (see
the monograph by Albeverio, Gesztezy, H~egh-Krohn, and Holden [AGHKH] and de
tailed bibliography therein). In the present monograph, we develop abstract approaches
to the study of singular perturbations based on the notion of quadratic forms.
V. Koshmanenko, Singular Quadratic Forms in Perturbation Theory
© Springer Science+Business Media Dordrecht 1999
2 Introduction
In general, there are many ways of taking into account the effect of singular
perturbation. The method of quadratic forms, the method of self-adjoint extensions, and
the method of orthogonal extensions are among the most extensively used methods of
this kind. They are applied and developed in the present monograph.
We now briefly describe the essence of each of these methods.
Assume that A is a positive self-adjoint operator (the Hamiltonian of free physical
system) with the domain of definition V(A) given in a Hilbert space Ji. Its perturba
tion is defined as a singular Hermitian quadratic form y with the domain of definition
n
Q (y). The singularity of y means that the set V = (Ker y) V(A) is dense in Ji. It is
required to construct and investigate the properties of the self-adjoint operator Ay that
corresponds to the Hamiltonian of the perturbed system. It is natural that the operators A
and Ay must coincide in V, i.e.,
Consequently, if Ay exists, then it is a self-adjoint extension of the symmetric operator
Thus, Ay"# A + V for any closed linear operator V in Ji because, otherwise, V must
be equal to zero in the dense set V.
The choice of methods used for the construction of the perturbed operator Ay as well
as the possibility of this construction depend on the degree of singularity of the
perturbation y. A natural classification of the degrees of singularity of the quadratic
form y as a perturbation of the operator A appears in the scale of Hilbert spaces
associated with A (A-scale)
where Jik = V(A k/2) in the graph norm. In the present monograph, we do not consider
the case of regular perturbations, where the form y is densely defined, bounded from
below, and closed in Ji from the domain V(A)n Q(y).
We say that a form y determines a perturbation from the Ji_, (A)-class if it is
bounded from below and closed in the space Ji, and, moreover, if the set V =
n
(Kery) V(A) is dense in Ji. In this case, y generates a self-adjoint operator V that
acts from Ji, into Ji_,.
In the present monograph, we consider positive forms and forms of finite ranks.
For perturbations from the class Ji_, (A), it is customary to use the method of quad
ratic forms, which is fairly well developed and have many applications (see, e.g.,
[AGHKH, AGS, Ber3, BeK, Chu, DaS, DeO, Fra, GeS, G11, KiS, KosS, KosI3, KosI6,
KosI8-Kos22, LyM, Mik, Nen3, Seg, Shal, Sha2, SimI-SimS, StovI, Stov2, Tip, and
