Stochastic Analysis and Mathematical Physics: ANESTOC ’98 Proceedings of the Third International Workshop

167 Pages·2000·6.812 MB·English

by  Raffaella Carbone (auth.), Rolando Rebolledo (eds.)

Trends in Mathematics
Trends in Mathematics is a book series devoted to focused collections of articles arising
from conferences, workshops or series of lectures.
Topics in a volume may concentrate on a particular area of mathematics, or may
encompass a broad range of related subject matter. The purpose of this series is both
progressive and archival, a context in which to make current developments available
rapidly to the community as well as to embed them in a recognizable and accessible way.
Volumes of TIMS must be of high scientific qUality. Articles without proofs, or which
do not contain significantly new results, are not appropriate. High quality survey papers,
however, are welcome. Contributions must be submitted to peer review in a process
that emulates the best journal procedures, and must be edited for correct use of language.
As a rule, the language will be English, but selective exceptions may be made. Articles
should conform to the highest standards of bibliographic reference and attribution.
The organizers or editors of each volume are expected to deliver manuscripts in a form
that is essentially "ready for reproduction." It is preferable that papers be submitted in
one of the various forms of TEX in order to achieve a uniform and readable appearance.
Ideally, volumes should not exceed 350-400 pages in length.
Proposals to the Publisher are welcomed at either:
Birkhauser Boston, 675 Massachusetts Avenue, Cambridge, MA 02139, U.S.A.
[email protected]
or
Birkhauser Verlag AG, PO Box 133, CH-40l0 Basel, Switzerland
[email protected]
Stochastic Analysis and
Mathematical Physics
ANESTOC '98
Proceedings of the Third International Workshop
Rolando Rebolledo
Editor
Springer Science+Business Media, LLC
Rolando Rebolledo
Facultad de Matemâticas
Pontificia Universidad Catolica de Chile
Casila 306, Santiago 22
Chile
Library of Congress Cataloging.in·Publication Data
International Workshop on Stochastic Analysis and Mathematical Physics (3rd: 1998 :
Santiago, Chile)
Stochastic analysis and mathematical physics : ANESTOC '98: proceedings ofthe third
international workshop / Rolando Rebolledo, editor.
p. cm.-(Trends in mathematics)
Includes bibliographical references.
ISBN 978-1-4612-7118-5 ISBN 978-1-4612-1372-7 (eBook)
DOI 10.1007/978-1-4612-1372-7
1. Stochastic analysis-Congresses. 2. Mathematical physics-Congresses. 1.
Rebolledo, Rolando, 1947- II. Title. III. Series.
QC20.7.S8158 1998
519.2-dc21 00-036050
CIP
AMS Subject Classifications: 60H30, 60H99, 6OJ99, 81S05, 8lU99
Printed on acid-free paper.
© 2000 Springer Science+Business Media New York
Originally published by Birkhăuser Boston in 2000
Softcover reprint of the hardcover 1 st edition 2000
AU rigbts reserved. This work may not be translated or copied in whole or in part without the written
pennission of the publisher Springer Science+Business Media, LLC, except for brief excerpts in
connection with reviews or scholarly analysis. Use in connection with any form ofinformation storage
and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now
known or hereafter developed is forbidden.
The use of general descriptive narnes, trade names, trademarks, etc., in this publication, even if the
former are not especiaUy identified, is not to be taken as a sign that such narnes, as understood by the
Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
ISBN 978-1-4612-7118-5 SPIN 10768765
Typeset by the editor in ~T#.
9 8 7 6 543 2 1
Contents
Chapter 1. Exponential L2-Convergence of Some Quantum Markov
Semigroups Related to Birth-and-Death Processes
Raffaella Carbone ................................................... 1
Chapter 2. Conservativity of Quantum Dynamical Evolution Systems
A.M. Chebotarev, J. C. Garda, and R. Quezada ..................... 23
Chapter 3. Upper Bounds on Bogolubov's Inner Product: Quantum
Systems of Anharmonic Oscillators
M. Corgini .......................................................... 33
Chapter 4. Bernstein Processes Associated with a Markov Process
A.B. Cruzeiro, Liming Wu, and J.C. Zambrini ...................... 41
Chapter 5. A Simple Singular Quantum Markov Semigroup
Franco Fagnola ..................................................... 73
Chapter 6. On a Theory of Resonance in Quantum
Mechanical Scattering
Claudio Fernandez and Kalyan B. Sinha ............................ 89
Chapter 7. Representation of the q-Deformed Oscillator
Alain Guichardet .................................................... 97
Chapter 8. On the Existence of Exponentials of Quadratic
Polynomials of Field Operators on Fock Space
Erik B. Nielsen and Ole Rask ...................................... 101
Chapter 9. The Wave Map of Feller Semigroups
Rolando Rebolledo ................................................. 109
Chapter 10. On the Korovkin Property and Feller Semigroups
Jan A. Van Casteren .............................................. 123
Chapter 11. An Example of the Singular Coupling Limit
Wilhelm von Waldenfels ........................................... 155
Preface
The seminar on Stochastic Analysis and Mathematical Physics started in
1984 at the Catholic University of Chile in Santiago and has been an on
going research activity. Since 1995, the group has organized international
workshops as a way of promoting a broader dialogue among experts in the
areas of classical and quantum stochastic analysis, mathematical physics
and physics.
This volume, consisting primarily of contributions to the Third Inter
national Workshop on Stochastic Analysis and Mathematical Physics (in
Spanish ANESTOC), held in Santiago, Chile, in October 1998, focuses on
an analysis of quantum dynamics and related problems in probability the
ory. Various articles investigate quantum dynamical semigroups and new
results on q-deformed oscillator algebras, while others examine the appli
cation of classical stochastic processes in quantum modeling.
As in previous workshops, the topic of quantum flows and semigroups
occupied an important place. In her paper, R. Carbone uses a spectral type
analysis to obtain exponential rates of convergence towards the equilibrium
of a quantum dynamical semigroup in the £2 sense. The method is illus
trated with a quantum extension of a classical birth and death process.
Quantum extensions of classical Markov processes lead to subtle problems
of domains. This is in particular illustrated by F. Fagnola, who presents a
pathological example of a semigroup for which the largest * -subalgebra (of
the von Neumann algebra of bounded linear operators of £2 (lR+, IC)), con
tained in the domain of its infinitesimal generator, is not a-weakly dense.
The extension of Markov theory to a non-commutative framework poses
another fundamental problem, that of preserving the unit of the algebra by
the action of a given quantum dynamical semigroup (the so-called con
servativity property). In their paper, A. Chebotarev, J. Garda and R.
Quezada provide a criterion for conservativity of minimal solutions of time
dependent master equations with unbounded coefficients.
The asymptotic analysis of quantum dynamical semigroups are repre
sented in two papers: that of W. von Waldenfels who provides an example
of the singular coupling limit, and that of C. Fernandez and KB. Sinha
who analyze quantum mechanical resonance.
The theme of quantum oscillators appears in two articles. M. Corgini
viii Preface
concentrates on studying systems of quantum anharmonic oscillators. He
obtains upper bounds for the so-called Bogolubov's inner product for cre
ating and annihilating Bose operators appearing in the above systems.
A. Guichardet provides a representation of the q-deformed oscillator al
gebra.
Fock spaces are the keystone in the development of quantum stochastic
analysis. In this volume, E.B. Nielsen and O. Rask provide an explicit
expression for the exponential of quadratic polynomials of field operators
on Fock space.
Classical stochastic analysis is represented within this volume as well,
although connected with the "quantum way of life." Indeed, R. Rebolledo
and J. van Casteren write about Feller semigroups with two different aims.
Motivated by a discussion with K. B. Sinha, the first author presents an
example of a commutative wave map, a notion inspired from quantum
scattering and extended by Rebolledo in a previous paper to the theory of
quantum dynamical systems. J. van Casteren proves that operators that
satisfy a property due to Korovkin may be extended as generators of Feller
processes.
Finally, and continuing with the relationship of classical stochastic
analysis and quantum mechanics, A.B. Cruzeiro, Liming Wu and J.C.
Zambrini present a general description of Bernstein processes. This is a class
of diffusion processes, relevant to the probabilistic counterpart of quantum
theory known as Euclidean quantum mechanics.
On behalf of the organizers, I thank all participants of ANESTOC '98
for their interesting contributions and passionate discussions.
I gratefully acknowledge the support received from several grants and
institutions, namely the "Catedra Presidencial en Analisis Cualitativo de
Sistemas Dinamicos Cuanticos", "Direcci6n de Investigaci6n y Postgrado
Universidad Cat6lica" , FONDECYT Projects 1960917 and 1990439.
Rolando Rebolledo
December, 1999
Stochastic Analysis and
Mathematical Physics
Chapter 1
Exponential L2-Convergence
of Some Quantum Markov
Semigroups Related to
Birth-and-Death Processes
Raffaella Car bone
ABSTRACT Given a quantum Markov semigroup (1tk~o on B(h), with a
faithful normal invariant state p, we associate to it the semigroup (Tt)t>o on
Hilbert-Schmidt operators on h (the L2(p) space) defined by Tt(//4xpI74) =
//41t(X)//4. This allows us to use spectral theory to study the infinitesimal
generator of (Ttk~o and deduce information on the speed of convergence to equi
librium of the given semigroup. We apply this idea to show that some quantum
Markov semigroups related to birth-and-death processes converge to equilibrium
exponentially rapidly in L 2 (p) .
1 Introduction
Let B(h) be the von Neumann algebra of all bounded operators on a Hilbert
space h and let P(h) (with p E [1, coD be the Banach space of bounded
operators on h such that IxlP has finite trace endowed with the norm Ilxll =
(tr (lxIP))l/p. We denote by 1 the identity operator on h.
A quantum Markov semigroup (QMS) on B(h) is a w*-continuous semi
group ('Jt)t>o of linear operators on B(h) with the following properties:
1. 'Jt(l) = 1 for every t ~ 0,
2. the map 'Jt is completely positive for every t ~ 0,
3. the map x -+ 'Jt(x) is w*-continuous on B(h).
We refer to [6] for a detailed study of complete positivity. Here we recall
only that completely positive, identity preserving maps enjoy the so-called
R. Rebolledo (ed.), Stochastic Analysis and Mathematical Physics
© Springer Science+Business Media New York 2000
2 R. Carbone
Schwarz property
'Jt(x*)'Jt(x) :::; 'Jt(x*x)
for each x E 13(h).
QMS is the natural generalisation of a classical Markov semigroup. In
deed a positive linear map on a commutative C* or von Neumann algebra
is also completely positive (see, e.g., [10]).
QMSs were introduced in physics to model the decay to equilibrium of
quantum open systems. The equilibrium state is represented by a positive
operator p with trace 1 such that tr (p'Jt(x)) = tr (px) for every x E 13(h).
The decay to the equilibrium state p is expressed by limt--->oo tr (O''Jt(x)) =
tr (px) for every x E 13(h) and every positive operator 0' with trace 1.
In the applications it is sometimes implicitly assumed that the above
convergence occurs at an exponential speed. However, rigorous proofs of
this fact are often lacking.
The speed of convergence of classical Markov semigroups has been ex
tensively studied (see, for example, [8] and references therein). One of the
most powerful techniques consists in associating with a given Markov semi
group (St)t;:::o, on a space of bounded measurable functions over JRd, say,
with invariant density 7r, another semigroup (St)t>o on L2(JRd) defined by
(1.1.1)
This allows the use of spectral analysis to study the infinitesimal generator
of (St)t;:::o and give estimates of the speed of convergence to equilibrium of
the given semigroup (St)t;:::o.
In this paper, which is a part of a joint work with F. Fagnola, we study
the same problem for some QMS with a faithful normal invariant state p.
There are several non-commutative analogues of the embedding f ~ 7r1/2 f
of bounded measurable functions over JRd in L2(JRd): the right (resp. left)
embedding x ~ Xpl/2 (resp. x ~ pl/2X), the symmetric embedding x ~
pl/4xpl/4, and so on. Here we shall use the symmetric one, which seems
easier to handle.
We consider the QMS on 13( h), h being the Hilbert space [2 (N) of square
summable sequences of complex numbers, with infinitesimal generator
£(x) -21 {fl2(N)x - 2fl(N)SxS* fl(N) + Xfl2(N)}
-~ {>..2(N)x - 2>"(N)S*xS>"(N) + x>..2(N)} (1.1.2)
where Sand N are the shift and the number operators respectively and >..
and fl are two positive functions defined on N.
As shown in [4], the restriction of such an operator C to the commutative
algebra generated by N is the infinitesimal generator of a birth-and-death
process with infinitesimal rates >..2(n) and fl2(n). Necessary and sufficient