Diagonalizing Quadratic Bosonic Operators by Non-autonomous Flow Equations

134 Pages·2016·0.967 MB·English

by  Volker Bach, Jean-bernard Bru

MEMOIRS
of the
American Mathematical Society
Volume 240 • Number 1138 (fourth of 5 numbers) • March 2016
Diagonalizing Quadratic Bosonic
Operators by Non-Autonomous Flow
Equations
Volker Bach
Jean-Bernard Bru
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
MEMOIRS
of the
American Mathematical Society
Volume 240 • Number 1138 (fourth of 5 numbers) • March 2016
Diagonalizing Quadratic Bosonic
Operators by Non-Autonomous Flow
Equations
Volker Bach
Jean-Bernard Bru
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
Providence, Rhode Island
Library of Congress Cataloging-in-Publication Data
Names: Bach,Volker,1965–—Bru,J.-B.(Jean-Bernard),1973–
Title: Diagonalizing quadratic bosonic operators by non-autonomous flow equations / Volker
Bach,Jean-BernardBru.
Description: Providence,RhodeIsland: AmericanMathematicalSociety,2016. —Series: Mem-
oirsoftheAmericanMathematicalSociety,ISSN0065-9266;volume240,number1138—Includes
bibliographicalreferences.
Identifiers: LCCN2015045925—ISBN9781470417055(alk. paper)
Subjects: LCSH:Hamiltonianoperator. —Matrices. —Hilbertspace.
Classification: LCC QC174.17.H3 B33 2016 — DDC 515/.39–dc23 LC record available at
http://lccn.loc.gov/2015045925
DOI:http://dx.doi.org/10.1090/memo/1138
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Contents
Chapter I. Introduction 1
Chapter II. Diagonalization of Quadratic Boson Hamiltonians 3
II.1. Quadratic Boson Operators 3
II.2. Main Results 5
II.3. Historical Overview 8
Chapter III. Brocket–Wegner Flow for Quadratic Boson Operators 11
III.1. Setup of the Brocket–Wegner Flow 11
III.2. Mathematical Foundations of our Method 13
III.3. Asymptotic Properties of the Brocket–Wegner Flow 15
Chapter IV. Illustration of the Method 19
IV.1. The Brocket–Wegner Flow on Bogoliubov’s Example 19
IV.2. Blow–up of the Brocket–Wegner Flow 25
Chapter V. Technical Proofs on the One–Particle Hilbert Space 27
V.1. Well–Posedness of the Flow 27
V.2. Constants of Motion 54
V.3. Asymptotics Properties of the Flow 61
Chapter VI. Technical Proofs on the Boson Fock Space 75
VI.1. Existence and Uniqueness of the Unitary Propagator 75
VI.2. Brocket–Wegner Flow on Quadratic Boson Operators 85
VI.3. Quasi N–Diagonalization of Quadratic Boson Operators 92
VI.4. N–Diagonalization of Quadratic Boson Operators 95
Chapter VII. Appendix 101
VII.1. Non–Autonomous Evolution Equations on Banach Spaces 101
VII.2. Autonomous Generators of Bogoliubov Transformations 111
VII.3. Trace and Representation of Hilbert–Schmidt Operators 115
References 121
iii

Abstract
We study a non–autonomous, non-linear evolution equation on the space of
operators on a complex Hilbert space. We specify assumptions that ensure the
global existence of its solutions and allow us to derive its asymptotics at temporal
infinity. We demonstrate that these assumptions are optimal in a suitable sense
and more general than those used before. The evolution equation derives from the
Brocket–Wegner flow that was proposed to diagonalize matrices and operators by
a strongly continuous unitary flow. In fact, the solution of the non–linear flow
equation leads to a diagonalization of Hamiltonian operators in boson quantum
field theory which are quadratic in the field.
ReceivedbytheeditorSeptember2,2013and,inrevisedform,March18,2014.
ArticleelectronicallypublishedonNovember13,2015.
DOI:http://dx.doi.org/10.1090/memo/1138
2010 MathematicsSubjectClassification. Primary47D06,81Q10.
Key words and phrases. Quadratic operators, flow equations for operators, evolution equa-
tions,Brocket–Wegnerflow,doublebracketflow.
Volker Bach, Institut fu¨r Analysis und Algebra, TU Braunschweig, Pockelsstraße 11, 38106
Braunschweig.
Jean-Bernard Bru, Departamento de Matem´aticas, Facultad de Ciencia y Tecnolog´ıa, Uni-
versidaddelPa´ısVasco,Apartado644,48080Bilbao–and–BCAM-BasqueCenterforApplied
Mathematics,Mazarredo,14. 48009Bilbao –and–IKERBASQUE,BasqueFoundation forSci-
ence,48011,Bilbao.
(cid:2)c2015 American Mathematical Society
v

CHAPTER I
Introduction
In theoretical physics, the second quantization formalism is crucial to treat
many–particle problems. In the first quantization formalism, i.e., in the canonical
ensemble, the number of particles of the corresponding wave function stays fixed.
Whereas in second quantization, i.e., in the grand–canonical ensemble, the particle
number is not fixed and their boson or fermion statistics is incorporated in the
well–known creation/annihilation operators acting on Fock space. Within this lat-
ter framework, we are interested in boson systems, the simplest being the perfect
Bosegas,i.e.,asystemwithnointeractiondefinedbyaparticlenumber–conserving
quadratic Hamiltonian. By quadratic Hamiltonians we refer to self–adjoint oper-
ators which are quadratic in the creation and annihilation operators. It is known
since Bogoliubov and his celebrated theory of superfluidity [1] that such quadratic
operatorsarereducedtoaperfectgasbyasuitableunitarytransformation,seealso
[2, Appendix B.2]. See also [32] with discussions of [33] in the finite dimensional
case.
Diagonalizations of quadratic boson operators are generally not trivial, and in
this paper, we investigate this question under weaker conditions as before. Indeed,
after Bogoliubov with his u–v unitary transformation [1,2], general results on
this problem have been obtained for real quadratic operators with bounded one–
particle spectrums by Friedrichs [3, Part V], Berezin [4, Theorem 8.1], Kato and
Mugibayashi [5, Theorem 2]. In the present paper we generalize previous analyses
ofreal quadraticboson Hamiltonians withpositive one–particle spectrumbounded
above and below away from zero to complex, unbounded one–particle operators
withoutagapabovezero. Thisgeneralizationisobviouslyimportantsince,formost
physically interesting applications, the one–particle spectrum is neither bounded
above nor bounded away from zero. Moreover, our proof is completely different.
Its mathematical novelty lies in the use of non–autonomous evolution equations as
a key ingredient.
More specifically, we employ the Brocket–Wegner flow, originally proposed by
Brockettforsymmetricmatricesin1991[6]and,inadifferentvariant,byWegnerin
1994forself–adjointoperators[7]. Thisflowleadstounitarilyequivalentoperators
via a non–autonomous hyperbolic evolution equation. The mathematical founda-
tion of such flows [6,7] has recently been given in [8]. Unfortunately, the results
[8] do not apply to the model(cid:2)s. In this p(cid:3)aper we prove the well–posedness of the
Brocket–Wegner flow ∂ H = H ,[H ,N] , where for t ≥ 0, [H ,N] := H N−NH
t t t t t t t
is the commutator of a quadratic Hamiltonian H and the particle number opera-
t
tor N acting on the boson Fock space. Establishing well–posedness is non–trivial
here because the Brocket–Wegner flow is a (quadratically) non–linear first–order
differential equation for unbounded operators. It is solved by an auxiliary non–
autonomous parabolic evolution equation.
1
2 I.INTRODUCTION
Indeed, non–autonomous evolution equations turn out to be crucial at two
different stages of our proof:
(a) T(cid:2)o show for(cid:3)t≥0 the well–posedness of the Brocket–Wegner flow ∂tHt =
H ,[H ,N] for a quadratic operator H via an auxiliary system of non–
t t 0
linear first–order differential equations for operators;
(b) To rigorously define a family of unitarily equivalent quadratic operators
H =U H U∗ as a consequence of the Brocket–Wegner flow.
t t 0 t
To be more specific, (a) uses the theory of non–autonomous parabolic evolution
equations via an auxiliary systems of non–linear first–order differential equation
for operators. Whereas the second step (b) uses the theory of non–autonomous
hyperbolic evolution equations to define the unitary operator U by U := 1 and
t 0
∂ U = −iG U , for all t > 0, with generator G := i[N,H ]. For bounded genera-
t t t t t t
tors,theexistence,uniquenessandevenanexplicitformoftheirsolutionisgivenby
the Dyson series, as it is well–known. It is much more delicate for unbounded gen-
erators, which is what we are dealing with here. It has been studied, after the first
resultofKatoin1953[9],fordecadesbymanyauthors(Katoagain[10,11]butalso
Yosida,Tanabe,Kisynski,Hackman,Kobayasi,Ishii,Goldstein,Acquistapace,Ter-
reni, Nickel, Schnaubelt, Caps, Tanaka, Zagrebnov, Neidhardt), see, e.g., [12–16]
and the corresponding references cited therein. Yet, no unified theory of such lin-
ear evolution equations that gives a complete characterization analogously to the
Hille–Yosidagenerationtheoremsisknown. ByusingtheYosidaapproximation,we
simplifyIshii’sproof[17,18]andobtainthewell–posednessofthisCauchyproblem
in the hyperbolic case in order to define the unitary operator U .
t
Next, by taking the limit t → ∞ of unitarily equivalent quadratic operators
H =U H U∗,undersuitableconditionsonH ,wedemonstratethatthelimitoper-
t t 0 t 0
atorH∞ isalsounitarilyequivalenttoH0. Thisissimilar toscatteringinquantum
field theory since we analyze the strong limit U∞ of the unitary operator Ut, as
t → ∞. The limit operator H∞ = U∞H0U∗∞ is a quadratic boson Hamiltonian
which commutes with the particle number operator N, i.e., [H∞,N] = 0 – a fact
which we refer to as H∞ being N–diagonal. In particular, it can be diagonalized
by a unitary on the one–particle Hilbert space, only. Consequently, we provide in
this paper a new mathematical application of evolution equations as well as some
general results on quadratic operators, which are also interesting for mathematical
physicists.
The paper is structured as follows. In Section II, we present our results and
discuss them in the context of previously known facts. Chapter III contains a
guidelinetoourapproachintermsoftheorems, whereasSectionIVillustratesiton
an explicit and concrete case, showing, in particular, that a pathological behavior
of the Brocket–Wegner flow is not merely a possibility, but does occur. Sections
V–VI are the core of our paper, as all important proofs are given here. Finally,
Section VII is an appendix with a detailed analysis in Section VII.1 of evolution
equationsforunbounded operatorsofhyperbolic typeonBanach spaces, andwith,
in Sections VII.2 and VII.3, some comments on Bogoliubov transformations and
Hilbert–Schmidt operators. In particular, we clarify in Section VII.1 Ishii’s ap-
proach [17,18] to non–autonomous hyperbolic evolution equations.
CHAPTER II
Diagonalization of Quadratic Boson Hamiltonians
In this chapter we describe our main results on quadratic boson Hamiltonians.
First, we define quadratic operators in Section II.1 and present our findings in
Section II.2 without proofs. The latter is sketched in Chapter III and given in full
detail in Chapters V–VI. Section II.3 is devoted to a historical overview on the
diagonalization of quadratic operators.
II.1. Quadratic Boson Operators
To fix notation, let h := L2(M) be a separable complex Hilbert space which
we assume to be realized as a space of square–integrable functions on a measure
space (M,a). The scalar product on h is given by
(cid:4)
(II.1) (cid:6)f|g(cid:7):= f(x)g(x)da(x) .
M
For f ∈h, we define its complex conjugate f¯∈ h by f¯(x) := f(x), for all x ∈ M.
For any bounded (linear) operator X on h, we define its transpose Xt and its
complex conjugate X¯ by (cid:6)f|Xtg(cid:7) := (cid:6)g¯|Xf¯(cid:7) and (cid:6)f|X¯g(cid:7) := (cid:6)f¯|Xg¯(cid:7) for f,g ∈ h,
respectively. Note that the adjoint of the operator X equals X∗ = Xt = Xt,
where it exists. The Banach space of bounded operators acting on h is denoted by
B(h), whereas L1(h) and L2(h) are the spaces of trace–class and Hilbert–Schmidt
operators, respectively. Norms in L1(h) and L2(h) are respectively denoted by
(II.2) (cid:9)X(cid:9) :=tr(|X|) , for X ∈L1(h) ,
1
and
(cid:5)
(II.3) (cid:9)X(cid:9) := tr(X∗X) , for X ∈L2(h) .
2
Note that, if there exists a constant K<∞ such that
(cid:6) (cid:6)
(cid:6)(cid:7)∞ (cid:6)
(cid:6) (cid:6)
(II.4) (cid:6) (cid:6)η |Xψ (cid:7)(cid:6)≤K ,
(cid:6) k k (cid:6)
k=1
for all orthonormal bases {η }∞ ,{ψ }∞ ⊆h, then
k k=1 k k=1
(cid:7)∞
(II.5) tr(X)= (cid:6)ϕ |Xϕ (cid:7)
k k
k=1
for any orthonormal basis {ϕ }∞ ⊆ h. Finally, we denote by 1 the identity
k k=1
operator on various spaces. Assume now the following conditions:
A1: Ω =Ω∗ ≥0 is a positive operator on h.
0 0
A2: B =Bt ∈L2(h) is a (non–zero) Hilbert–Schmidt operator.
0 0
3