C 0-Groups, Commutator Methods and Spectral Theory of N-Body Hamiltonians

473 Pages·1996·11.884 MB·English

by  Werner O. Amrein, Anne Boutet de Monvel, Vladimir Georgescu

Progress in Mathematics
Volume 135
Series Editors
H. Bass
J. Oesterle
A. Weinstein
Werner O. Amrein
Anne Boutet de Manvel
Vladimir Georgescu
Co·Groups,
Commutator Methods
and Spectral Theory of
N-Body Hamiltonians
Springer Basel AG
Authors:
Werner O. Amrein
Ecole de Physique
Universite de Geneve
24 Quai Ernest-Ansermet
1211 Geneve 4
Switzerland
Anne Boutet de Monvel and
Vladimir Georgescu
Institut de Mathematiques de Paris-lussieu
C.N.R.S. UMR 9994
Universite Paris VII Denis Diderot
UF.R. de Mathematiques
Case 7012
2, Place lussieu
75251 Paris Cedex 05
France
A CIP catalogue record for this book is available from the Library of Congress,
Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Amrein, Werner 0.: CO-groups, commutator methods and spectral theory of N-body
Hamiltonians / Werner o. Amrein; Anne Boutet de Monvel ; Vladimir Georgescu.
Basel ; Boston ; Berlin : Birkhauser, 1996
(Progress in mathematics ; Vol. 135)
NE: Boutet de Monvel, Anne:; Georgescu, Vladimir:; GT
1991 Mathematics Subject Classification 46L60.
This work is subject to copyright. All rights are reserved, whether the whole or part
of the material is concerned, specifically the rights of translation, reprinting, re-use
of illustrations, broadcasting, reproduction on microfilms or in other ways, and
storage in data banks. For any kind of use whatsoever, permission of the copyright
owner must be obtained.
© 1996 Springer Basel AG
Originally published by Birkhiiuser Verlag in 1996.
Softcover reprint of the hardcover I st edition 1996
Printed on acid-free paper produced of chlorine-free pulp. TCF 00
ISBN 978-3-0348-7764-0 ISBN 978-3-0348-7762-6 (eBook)
DOl 10.1007/978-3-0348-7762-6
987654321
Contents
Preface.............................................................. ix
Comments on notations ........................................... xiii
Chapter 1
Some Spaces of Functions and Distributions
1.1. Calculus on Euclidean Spaces .................................... 1
1.2. Distributions, Fourier Transforms ................................ 5
1.3. Estimates of Functions and their Fourier Transforms ............. 12
1.4. Rapidly Decreasing Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Chapter 2
Real Interpolation of Banach Spaces
2.l. Banach Spaces and Linear Operators 29
2.2. The K-Functional ............................................... 36
2.3. The Mean and the Trace Method ................................ 41
2.4. Comparison and Duality of Interpolation Spaces .................. 46
2.5. The Reiteration Theorem ........................................ 49
2.6. Interpolation of Operators ....................................... 52
2.7. Quasi-Linearizable Couples of B-Spaces .......................... 54
2.8. Friedrichs Couples ............................................... 59
vi Contents
Chapter 3
Co-Groups and Functional Calculi
3.1. Submultiplicative Functions and Algebras Associated to them 75
3.2. Co-Groups: Continuity Properties and Elementary Functional
Calculus ......................................................... 85
3.3. The Discrete Sobolev Scale Associated to a Co-Group ............ 94
3.4. Besov Spaces Associated to a Co-Group .......................... 122
3.5. Littlewood-Paley Estimates ...................................... 136
3.6. Polynomially Bounded Co-Groups ................................ 148
3.7. Co-Groups in Hilbert Spaces 160
Chapter 4
Some Examples of Co-Groups
4.1. Weighted Sobolev and Besov Spaces 171
4.2. Co-Groups Associated to Vector Fields 178
Chapter 5
Automorphisms Associated to Co-Representations
5.1. Regularity and Commutators .................................... 193
5.2. Regularity of Fractional Order ................................... 199
5.3. Regularity Preserving and Regularity Improving Operators ....... 209
5.4. The spaces .,a;,p(lRn) ............................................. 216
5.5. Commutator Expansions ......................................... 222
5.A. Appendix: Differentiability Properties of Operator-Valued
Functions ........................................................ 231
Chapter 6
Unitary Representations and Regularity
6.1. Remarks on the Functional Calculus for Self-adjoint Operators 236
6.2. Regularity of Self-adjoint Operators with respect to
Unitary Co-Groups .............................................. 242
6.3. Unitary Groups in Friedrichs Couples ............................ 253
6.4. Estimates on cp(Hd - cp(H2) ..................................... 260
6.A. Appendix: Remarks on the Functional Calculus Associated
to !.i)f in B (,yt?) .................................................. 264
Contents vii
Chapter 7
The Conjugate Operator Method
7.1. Locally Smooth Operators and Boundary Values of
the Resolvent .................................................... 272
7.2. The Mourre Estimate.................... ........................ 287
7.3. The Method of Differential Inequalities. . . . . . . . . . . . . . . . . . . . . . . . . .. 299
7.4. Self-adjoint Operators with a Spectral Gap ....................... 308
7.5. Hamiltonians Associated to Symmetric Operators in
Friedrichs Couples ............................................... 312
7.6. The Limiting Absorption Principle for Some Classes of
Pseudo differential Operators ..................................... 330
7.A. Appendix: The Gronwall Lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 349
7.B. Appendix: A Counterexample. Optimality of the Results on
the Limiting Absorption Principle ................................ 350
7.C. Appendix: Asymptotic Velocity for H = h(P) .................... 355
Chapter 8
An Algebraic Framework for the Many-Body Problem
8.1. Self-adjoint Operators Affiliated to C*-Algebras .................. 359
8.2. Tensor Products ................................................. 372
8.3. Q-Functions in a C*-Algebra Setting.............................. 380
8.4. Graded C*-Algebras ............................................. 391
Chapter 9
Spectral Theory of N-Body Hamiltonians
9.1. Tensorial Factorizations of ,Jt(X) ................................ 401
9.2. Semicompact Operators... ............... ........................ 403
9.3. The N-Body Algebra ............................................ 409
9.4. Non-Relativistic N-Body Hamiltonians .......................... . 414
9.A. Appendix: Remarks on the ~1,1 Property 430
Chapter 10
Quantum-Mechanical N-Body Systems
10.1. Clustering of Particles ........................................... 433
10.2. Quantum-Mechanical N-Body Hamiltonians. . . . . . . . . . . . . . . . . . . . .. 439
Bibliography ........................................................ 445
Notations ........................................................... 453
Index................................................................ 457
To our daughters
Vera, Violaine, Sonia, Tiphaine
Koyu, Ie religieux, dit:
seule une personne de compnihension reduite
desire arranger les choses en series completes.
C'est l'incompletude qui est desirable.
En tout, mauvaise est la regula rite.
Dans les palais d'autrefois, on laissait toujours
un bdtiment inacheve, obhgatoirement.
(Tsuredzurc Gusa, par Yoshida No Kancyoshi, XIVeme sieclc)
Preface
The relevance of commutator methods in spectral and scattering theory has
been known for a long time, and numerous interesting results have been ob
tained by such methods. The reader may find a description and references in the
books by Putnam [Pu], Reed-Simon [RS] and Baumgartel-Wollenberg [BW] for
example. A new point of view emerged around 1979 with the work of E. Mourre
in which the method of locally conjugate operators was introduced. His idea
proved to be remarkably fruitful in establishing detailed spectral properties of
N-body Hamiltonians. A problem that was considered extremely difficult be
fore that time, the proof of the absence of a singularly continuous spectrum
for such operators, was then solved in a rather straightforward manner (by
E. Mourre himself for N = 3 and by P. Perry, 1. Sigal and B. Simon for general
N). The Mourre estimate, which is the main input of the method, also has
consequences concerning the behaviour of N-body systems at large times. A
deeper study of such propagation properties allowed 1. Sigal and A. Soffer in
1985 to prove existence and completeness of wave operators for N-body systems
with short range interactions without implicit conditions on the potentials (for
N = 3, similar results were obtained before by means of purely time-dependent
methods by V. Enss and by K. Sinha, M. Krishna and P. Muthuramalingam).
Our interest in commutator methods was raised by the major achievements
mentioned above. In studying these papers we arrived at the conviction that
the field of applications of the method of locally conjugate operators was by no
means exhausted and also that the theory itself could be improved on an ab
stract level such as to cover most of the known results in spectral and scattering
theory and to obtain these results under sharper and more natural conditions.
The present monograph is a presentation of the principal outcomes of our efforts
in this direction.
It turned out that, in order to arrive at the refined version of the locally con
jugate operator method we were looking for, we had to have recourse to certain
non-Hilbertian techniques which are rarely used in spectral and scattering the
ory, such as real interpolation theory and Co-groups of automorphisms of C*-