Positive Commutator Method in Non-Equilibrium Statistical ... · Abstract Positive Commutator...

Positive Commutator Method in Non-Equilibrium Statistical Mechanics

.A thesis subrni t ted in conformity with the requirements

for the degree of Doctor of Philosophy

Graduate Department of Mathematics

University of Toronto

@ Copyright by Marco Merkli (2000)

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Abstract

Positive Commutator Method in Non-Equilibrium St at ist ical Mechanics

Ph.D. 2000

Marco Merkli

Department of Mathematics

University of Toronto

The main goal of this thesis is to develop the method of positive commu-

tators in the contest of quantum statistical mechanics.

Over the last twenty years. this method has been powering progress in

spectral analysis of Hamiltonians of quantum mechanical systems. It has re-

cently been applied to the problem of radiation. which is formulated in terms

of quantum field theory.

In this work. we estend the applicability of the positive commutator

method to positive temperature quantum field theory. More precisely. we

study an atom (an .V-level system) interacting with an infinitely extended

photon-field ( a massless Bose field) at temperature T > 0. In a GKS rep-

resentation of the C'-algebra of (quasi-local) observables for the quantum

system in question. the dynamics of the system is generated by a Liouville

operator acting on a positive temperature Hilbert space. Many key prop-

erties of the system. such as return to equilibrium (which is the fact that a

system perturbed from its equilibrium state converges back to it as time goes

to infinity). can be expressed in terms of the spectral characteristics of this

operator.

We apply the positive commutator method to the Liouville operators of

systems in question. Using this method. we obtain rather detailed spectral

information about these operators. This allows us to recover. with a partial

improvement. a recent fundamental result by several authors on return to

equilibrium for systems under consideration.

dedichi a Poppa e Bap

Acknowledgments

Above all. I thank my supervisor Professor 1.41. Sigal for having intro-

duced me to the world of mathematical physics. With his help. I have learned

much more than can be writ ten down in a thesis.

I am very grateful to Xancy for being with me. On many occasions in

our lives as graduate students. Yancy has shown me concretely how to reach

return to equilibrium.

Coming to Toronto for graduate studies has been an extremely pleasant

experience. I thank Canada and especially the Slathematics Department of

the University of Toronto for their warm welcome antl support. Many thanks

go to Nadia. Pat. Annette. Sliranda and Marie. antl especially to Ida for all

her help.

I gratefully acknowledge the financial support of the Mathematics Depart-

ment of the University of Toronto. the Government of Ontario and SSERC.

During the last stages of this work. I had the occasion to stay at the IHES

in Bures and the ETH in Ziirich. I would like to thank Professor 3. Frohlich

for his hospitality at the ETHZ and for enlightening discussions.

I end this highly non-erhaus t ive list of acknowledgments by thanking

Xetle and Ingy for their kind hospitality in Winterthnr.

Contents

1 Introduction and Main Results 1

. . . . . . . . . . . . . . . . . . . 1.1 Introduction to the problem 2

1.1.1 Mathematical description . Positive Comniutator tech-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . nique 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The model S

. . . . . . . . . . . . 1.2.1 .A class of open quant urn systems S

. . . . . . . . . . 1.2.2 Yonrelat ivistic matter and radiation 12

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Main results 15

. . . . . . . . . . . . . . . . . . . . 1.4 Review of previous results 20

. . . . . . . . . . . . . . . . . . . . 1.5 Organization of the thesis 24

2 Mathematical framework 26

. . . . . . . . . . . . . . . . . . . 2.1 Quantum statistical systems 26

. . . . . . . . . . . . . . . . . . 2.1.1 Observables and states 26

2.1.2 Representations of the algebra of observables and the

. . . . . . . . . . . . . . . . . . . . . . . . . Liouvillian 2s

. . . . . . . . . . . . . . 2.1.3 Positive temperature systems 30

. . . . 2.1.4 Return to equilibrium . spectral characterization 33

. . . . . . . 2.2 The particlefield system at positive temperature 36

. . . . . . . . . . . . . . . 2.2.1 The non-interac t ing system 36

. . . . . . . . . . . . . . . . . . 2.2.2 The interacting system 40

2.2.3 JakSiC-Pillet gluing . unitarily transformed Liouvillian . selfadjointness . . . . . . . . . . . . . . . . . . . . . . . 43

3 Proof of Theorem 1.1 52

3.1 Positive commutator with respect to spectral localization in Lo 52

3.1.1 The Feshbach method . . . . . . . . . . . . . . . . . . 56

. . . . . . . . . . . . . . . . . . 3.1.2 ProofofTheorem3.4. 59

3.2 Positive Commutator with respect to spectral localization in L 66

4 Proof of Theorem 1.2 75

4.1 The proof of Theorem 1.2 in the spirit of the Virial Theorem . 76

4.2 Perturt~ationofthe 1I;MSstatc . . . . . . . . . . . . . . . . . . 84

85

.4.1 Proof of Theorem 1.4 . . . . . . . . . . . . . . . . . . . . . . . 85

A.2 Proof of Proposition 3.1 . . . . . . . . . . . . . . . . . . . . . 90

A.3 Proof of Proposition 3.2 . . . . . . . . . . . . . . . . . . . . . 96

A.4 Proof of Proposition 3.S . . . . . . . . . . . . . . . . . . . . . 97

A.5 Proof of Remark 3 after Theorem 1.1 . . . . . . . . . . . . . . 102

A.6 Properties of some commutators . . . . . . . . . . . . . . . . . 104

.4 .7 On a smooth partition of unity . . . . . . . . . . . . . . . . . 112

115

B . 1 Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

B.2 Operator calculus . . . . . . . . . . . . . . . . . . . . . . . . . I20

Bibliography

vii

Chapter 1

Introduction and Main Results

One of the goals of this thesis is to estend the method of positive commuta-

tors. which was very success full^* applied to zero temperature problems. to

positive temperatures. i.e. to non-equilibrium statistical mechanics.

Using this technique. we give another proof of a basic property of large

quantum systems. known as return to equilibrium. We also extend an im-

portant Virial-Theorem type technique beyond its traditional range of appli-

cat ions.

Our main technical result is a positive commutator estimate (also called a

SIourre estimate) for the Liouville operator (or thermal Hamiltonian). which

is the time translation generator of the positive temperature system. This

result holds for a wider class of systems than previously considered.

There is a restriction on the class of systems for which we prove return to

equilibrium. due to the Virial-Theorem type result mentioned above. This is

the first result of this kind. and we expect that it will be improved to yield

the return to equilibrium result for a considerably wider class of systems.

1.1 Introduction to the problem

In this section. we give a general description of the systems and phenomena

w e are interested in. Before introducing the main mathematical problem

together with an outline of the strategy used to solve it in this work (see

subsection 1.1.1). let us describe the problem in an infornial way.

In this work. we study open quantum systems consisting of a small sub-

system interacting with a large environment.

.I typical example for the small system is an .V-body system of quantum

mechanical particles confined to a compact region in physical space. We call

the small system the particle sgstem.

The large system. also called the reservoir. or heat bath. consists of a spa-

tially infinitely estendeci massless bosonic field at a given temperature T > 0.

More precisely. the field is initially in an -asymptotic thermal equilibrium"

at temperature T > 0. This means. roughly speaking. that it is physically

indistinguishable from thermal equjlibrium states (KNS states) at the given

temperature in a neighbourhood of spatial infinity. Technically speaking, the

field is initially iu a state normal with respect to the IiSIS-state of the free

bosonic field at temperature T > 0.

Our main interest is the phenomenon of Return to Equilibrium (RTE). .A

system is said to exhibit RTE if an initial state which is an arbitrary asymp-

totic thermal equilibrium state at temperature T > 0 converges in the large

time limit to an equilibrium state (IiMS state) of the system at the same

temperature T. Among the family of initial conditions indicated here are

states where the particle system is in any initial state. and the field is in an

asymptotic thermal equilibrium state at temperature T.

The phenomenon of RTE can he mathematically formulate.' as a spec-

tral problem for the time-translation generator of the positive temperature

system. called the Lioaville operator. or thermal Hamiltonian. If it has no

eigenvalues. except for a simple one at zero. then the property of RTE follows.

In this work, we tackle the spectral analysis of the Liouville operator using

the positive commutator method. which tve extend from the zero temperature

to the positive temperat tire case.

1.1.1 Mat hematical description, Positive Commutator technique

Let us give a mathematical picture of the problem of RTE. including an our-

line of the strategy (called the Positive Cornnut at or ( PC ) technique ) we use

to show the spectral properties of the Liouville operator.

In order to make the mat hematical description more easily understand-

able. some preliminary esplanations are helpful. We refer to Chapter 2 for

more details.

A quantum statistical system is described by observables forming a C'-

algebra 3, and a one-parameter group of *-automorphisms at on II governing

the dynamics of the observables. States of the system are positive linear nor-

malized functionals on the algebra of observables. The evolution of a state ;2

is given by its composition with the dynamics: st = o at . .A state is called

station an^ if st = Y.

There is a distinguished finite dimensional class of stationary states (usu-

ally of dimension one or two). called the class of IiMS states (relative to the

given dynamics). One of the parameters of this class is called the (inverse)

temperature /? of the system. The KMS states are defined via an abstract

IiMS condition. and it follows in particular from this condition that the 1<MS

states are stable under small perturbations of the dynamics (apart from be-

ing stationary). This means that starting from a KlIS state m.r.t. a given

dynamics at. for a slightly perturbed dynamics. there is also a KSIS state.

The equilibrium states of a system at temperature 3 are by definition the

3-I<NS states.

The term equilibrium refers to dynamical stability in the sense that if w'

is an equilibrium state. then w' o at + as t + x;. for states i ~ ' that are

close to &t (as me will define below). This is called the property of return to

equilibrium. as it shows that a system perturbed from its equilibrium state

converges back to it in the large time limit. We point out that it is not known

generally that IiMS states exhibit this dynamical attractivity. The identifi-

cation of the equilibrium states with the K31S states can be justified so far

only by proving the property of return to equilibrium for concrete systems.

Our result is a step in this direction.

Using the GNS construction. one reformulates the setup in terms of a C'-

algebra described above in a Hilbert space framework. Yamely. given any

state J on 3. one can associate to it. via the GSS construction. a Hilbert

space 'H with a distinguished vector R and a representation x : 5% + D(R)

on the bounded operators on 31 s.t. s(n) = (R.;i(a)R). for a E 3. Here. (.:)

is the inner product in R. If the state is invariant under a t , then there is

a unique unitary group U ( t ) on 31 s.t. d ( a t ( a ) ) = (0. U m ( t ) n ( n ) L - ( t ) R ) and

U(t)R = Rl Vt. We say the dynamics at is unitarily implemented by i T ( t ). If

the unitary group L'( t ) is strongly continuous. then it has a selfadjoint gener-

ator L (Stone's theorem). called the Liouville operator. Notice that LR = 0.

Representing the dynamics of the system on a Hilbert space (as opposed to

a C'-algebra) allows us to connect the spectral properties of the Liouville

operator with the dynamical behaviour of the system, and spectral analysis

of selfadjoint operators is a vast subject with many tools available.

One shows that the Liouville operator constructed from a K M S state U:

(recall that I<MS states are stationary. i.e. invariant under the dynamics)

governs the evolution of states that are in a certain sense close to the K M S

state. called the normal states w.r.t. the given KMS state. and that the

relation between the spectrum of L and the dynamics of the normal states

implies: if the Liouville operator has no other eivenvalues than zero (recall

that there is always a zero eigenvector R). and if zero is moreover a simple

eigenvalue. then one has for any normal state

as t + x; and for any observable a. This is what w e understand by return

to equilibrium.

We are now ready to have a closer look at the class of Liouville operators

we will deal with. The Liouville operator consists of two parts.

L = L, + X I .

where La is the uncoupled Liouville operator. describing the two subsystems

(particles and field) when they do not interact. I is the interaction. and X is

a real (small) coupling parameter.

The uncoupled Liouville operator Lo is easy to analyze. Its spectrum

consists of a continuum covering the whole real axis. and it has embedded

eigenvalues. arranged symmetrically w.r. t . zero. Moreover. zero is a degen-

erate eigenvalue of Lo.

.According to the picture given above. we would like to show that for

X # 0: the spectrum of L has no eigenvalues. except for a simple one at zero.

.. a. . I .. 0 0

degenerate non-degenerate

In other words. me want to show that all nonzero eigendues of Lo are

unstable under the perturbation. and the perturbation removes the degen-

eracy of the zero eigenvalue. Quite generally. for the kind of systems we

consicler. one can construct a zero eigenfunction for L ( a IiMS state for the

coupled system) by using a perturbation series in X ( the zero-order term be-

ing the K M S state of the non-interacting system. this is called perturbation

of KMS states). This means that our task reduces to showing instability of

all nonzero eigentalues. and that the dimension of the nullspace of L is at

most one.

It is conventional wisdom that embedded eigenvalues are unstable un-

der generic perturbations. turning into resonances. Let us now outline the

technique we use to show instability of embedded eigenvalues: the Positive

Commutator (PC) technique.

To do so. we concentrate first on a nonzero (isolated) eigenvalue E of Lo

whose instability we want to show. The main idea is to construct an anti-

selfajoint operator A. called the adjoint operator. s. t. we have the following

PC estimate:

where 0 > 0 is a srictly positive number. Ea( L ) denotes the spectral projector

of L onto the interval 1. and [.. -1 is the commutator. Here. A contains the

eigenvalue e and no other eigenvalues of Lo. Equation ( 1.1) is also called

the (strict) Mowre estimate. If it is satisfied. then one shows that L has no

eigencalues in A by using the following argument by contradiction: suppose

that LC? = e ' k with e' E 1 and llc*11 = 1. Then we h a w EA(L)i? = ri*. and

the PC estimate (1.1) gives

where (.. .) denote the scalar product. On the other hand. formally espanding

the cornmu t ator yields

( t i l . [L. .-L]c*) = (v. [L - el. .-4]c*) = 2Re ((L - el)ci?. -4~7) = 0. (1.2)

which gives the contradiction 0 $ 0. hence showing that there cannot be any

eigenvalue of L in 1.

This formal proof is in general wrong. Indeed. both operators L and .-I

are unbounded. and one has to take great care of domain questions. including

the very definition of the commutator [L, -41.

Relation (1.2) i s called the Virial Theorem. and it can be made in many

concrete cases rigorous by approsimating the hypothetical eigenfunctim a. by "nice" vectors. The situation in which this works is quite generally given

in the case where [L,. A] is bounded relatively to L. which is in particular sat-

isfied for X-body Schrodinger systems. and systems of part ides coupled to a

field at zero temperature. However. in our case the condition is not satisfied,

and we have to develop a more general argument of this type.

The treatment of the zero eigenvalue is similar. except that we must not

try to prove ( 1.1 ) on the whole space Ran EA ( L ). since we know there is a

zero eigentalue. but only on Ran EA ( L ) P I . where P is the rank-one projector

onto the (known) eigenvector. and P' is its orthogonal complement.

1.2 Themodel

The goal of this section is to introduce the class of open quantum systems

we consider. see Section 1.2.1. In Section 12.2. we give the motivation for

the choice of this class. For a more detailed mathematical description. we

refer to Chapter 2.

From now on. we call the Liouville operator simply the Liouvillian.

1.2.1 A class of open quantum systems

We present the class of systems me will analyze and whose choice is motivated

by the quantum mechanical models of nonrelativistic matter coupled to the

radiation field (see Subsection 1.2.2). or matter interacting with a phonon

field (quantized modes of a lattice). or a generalized spin-boson system (see

e.g. [JPP] ). .A good review of physical models leading to the class of Hamil-

tonians considered here is found in [HSp].

The small system. also called the particle system. is described by a Hilbert

space R, as its state space. and by a Hamiltonian Hp which governs the

evolution. The Hamiltonian is a selfadjoint operator on R,. and we assume

that it has purely discrete spectrum:

with the further assumption that the eigenvalues EJ are finitely degenerate

and do not accumulate at any finite point. so that we have the finite trace:

We do not need to further specify the particle system. -1s a concrete esample.

one may think of a system of finitely many Schrodinger particles in a bos. or

a spin system. In some of our results (see Theorem 1.4 on the Fermi Golden

Rule Condition). me shall assume that the spectrum of Hp is finite (.V-level

system).

Yest. we describe the large system. also called the field. or the reservoir. It

is given by a spatially infinitely extended free massless bosonic field. Physical

esamples for the reservoir are the quantized elec trornagnet ic field (photons).

or the quantized vibrations of a lattice (phonons). In the case of the photons.

we are dealing with a field with two components (Coulomb gauge). while the

phonons are described by a scalar field. For notational sirnpicity. we take the

field to be scalar. i.e. the one particle space of the field is given by

and the state space of the field is the bosonic Fock space over fi:

where for n > 0. fi3:ym is the symmetrized n-fold tensor product of the

Hilbert space 4. and f jOf := @. This symmetrization takes into account

that we are dealing with Bosom. whose wave-functions are symmetric in the

arguments ( Bose-Einstein statistics).

The Harniltonian of the field is the second quantization of the multiplica-

tion operator by = Ik1. and if a m ( k ) . a(k) denote the (distribution valued)

creation and annihilation operators. then we can espress the Hamiltonian as

H, = d r ( s ) = $ku(lz)a-(k)a(k). 1 (1.5)

For details on creation and annihilation operators and on the second cpan-

tization. we refer to Appendix B. Section B.1.

The combined. uncoupled system at zero temperature is described by the

Hilbert space

R = H p 2 X j .

and its dynamics is determined by the selfadjoint Hamiltonian

H ~ = ~ , ~ n , + n , a ~ , .

We now describe the interaction between the small and the large system.

The full Hamiltonian of the interacting system is given by

H = Ho + X V . (1.6)

where the coupling constant X is a small real number. for notational conve-

nience and without loss of generality taken to be positive. and

Here. G is a bounded selfadjoint operator on Up. The function

called the form factor and the smoothed out creator is given by

We assume g to be a bounded CL-function. satisfying the following infra-red

(IR) and ultra-violet ( UV) conditions:

IR: Ig(P) I < CdP. for some p > 0. as -+ 0.

for some results. we assume p > 2.

UV: Ig(k)l 5 Cd-9. for some q > j /Z as, + m.

Here. we recall that J = / k ( . In addition, we assume that conditions (1.8)

hold for the derivative k g , if p. q are replaced by p - 1. q + 1.

Before we can state our main result, we need to introduce a condition on

the interaction, called the F e m i Golden Rule Condition. This is a condition

on the operator G and the function g ( k ) . as we explain i~elom. In the language

of quantum resonances. the Ferrni Golden Rule Condition expresses the fact

that the bifurcation of complex eigencalnes (resonance poles) of the deformed

Liouvillian takes place at second order in the perturbation (i.e. the lifetime

of the resonance is of the order A-' ).

We mention that the Liouvillian corresponding to the particle system at

positive temperature is given by L p = H,, 3 1 - 11 8::: H,. acting on the Hilbert

space R, 9 Rp (see (2.19) and the sentence after (2.27)). Thus L, has discrete

spec t n m given by

For every eigenvalue e of L,. we define an operator r ( e ) acting on the sub-

space Ran P( L, = e ) c 'H, :'; Up by

rnR(u. a ) P ( L p f e)6(Lp - E - u ) m ( u . a). P, XS'

(1.9)

where 8 denotes the Dirac function. and where the operator rn is given by

Here. gk.2 are two functions on IR x S2. constructed from g? see (2.25) and

(2.26). It is clear from (1.9) that r ( e ) is a non-negative selfadjoint operator.

The Fermi Golden Rule Condition is used to show instability of embedded

eigenvalues. For nonzero eigenvalues e # 0. the condition says that T(e) is

strictly positive:

for e f 0. 7, := inf o ( r ( e ) RanP(L, = E ) ) > 0. (1.11)

We will show that r ( 0 ) has a simple eigencalue at zero. the eigenvector

being the Gibbs state of the particle system. RP,. This reflects the fact that

the zero eigenmlue of Lo survives the perturbation. however. its degeneracy

is removed. i.e. the zero e igen~due of L is simple. The Fermi Golden Rule

Condition for E = 0 cannot read like (1.11). but we can have strict positivity

only on the cornple~nent of the zero eigenspace of r ( 0 ) . i.e. the Fermi Golden

Rule Condition for e = O is

yo := info (r(0) 1 Ran P( L, = 0) P&) > 0.

We give in Theorem 1.1 below explicit conditions on G and g(k) s.t. ( 1.11)

and (1.12) hold.

1.2.2 Nonrelativistic matter and radiation

In this subsection. we sum up some facts about the quantum mechanical sys-

tem of nonrelativistic matter and radiation (see also [BFS 1.21 ) and compare

it to the class of systems introduced in the previous subsection.

Xeglecting the vector nature of the photons (i.e. their two possible po-

larizations in the Coulomb gauge). the vector potential of the quantized

electromagnetic field is given at x E W3 (and at time zero) by

where ;(k) = 1k1 is the energy of a massless photon of momentum k. The

electric and magnetic fields are given by the time derivative and the curl of

the vector potential. respectively. The time evolution of -4 is given by the

Heisenberg evolution generated by the free field Hamiltonian. (1.5).

Yext. we describe the atom in the electromagnetic field. Its Hamiltonian

acts on the Hilbert space 31, :: R,. where RP is the .V-fold antisymmetrized

L2-space of Fermionic wave-functions. and it is given by

S

Hat = C(pJ - e.-l(r,))' + Coulomb potential. j = I

where we assume that the atom consists of .Y electrons (of mass 112) and

charge e (static nucleus). Here. p, = -iVJ is the momentum of the j-th

electron. and xJ is its position. The Coulomb potential takes care of the

elect ron-elec t ron repulsion am1 the elect ron-nucleus attract ion. Xotice that

we sum over the kinetic particle momenta pJ - e.-l(s,). (not the mere canon-

ical momenta pJ ) which comes from the fact that the particles are coupled

to the field via the electrostatic potential and the Lorentz force. We also ne-

glected the effect of the electron spin. i.e. we take the particles of the atom

to be spinless.

Next. in order to exhibit the perturbative character of the total Hamiito-

nian

one can perform a suitable unitary transformat ion acting on the particle

positions and field momenta (see [BFS 1 .S] 1 to obtain the following unitarily

equivalent Harniltonian for the total system:

Here. a = e 2 / c z 1/13: is the Feinstructure constant.

?jotice that the singularity / I 2 in the vector potential (1.13) causes

in many calculations problems (so-called infra-red problems. for 1 k 1 small).

In order to improve this infra-red behaviour. we perform the Pauli-Fierz

transfornation. which is a unitary transformation implemented by

Zi

erp (-iT.410)

and for a suitable r E R. the unitarily transformed Hamiltonian reads

H = C p,? + Coulomb potential + Hj

+ 2 5 1 / 2 ~ 3 1 2 E (0) I] + higher order terms.

where the higher order terms contain a factor am with rn > 312. and E ( 0 ) =

&.4(0. t ) is the electric field at the origin. In the higher order terms appear

powers of the particle positions. so if the atom is localized around zero. these

terms are small if a is small. Neglecting these terms constitutes the dipole

approximation.

Using the relation ( 1 .l3) to express E ( 0 ) in terms of the vector potential.

and neglecting the higher order terms. one obtains

H = Ho f Xc.

where Ho is the sum of the Hamiltonian of the particle system (not coupled

to the field).

constant, and

plus the free field Harniltonian. X = d l 2 is a small coupling

v is of the form

where G(k) is an operator acting on the particle Hilbert space and has the

following infra-red behaviour:

.An abstraction of this interaction. given by setting

where G is a hounded operator on the particle Hilbert space. and g is a

function with IR behaviour

yields our interaction (1.7). We point out that our methods also work for

interactions of the form (1.11). where G(k) is not of the special product form

( 1.15). The choice of our interaction (1.7) in this respect is made purely for

no tat ional convenience.

1.3 Main results

Our results concern the Liouvillian L at inverse temperature 3 = 1/T < cc. corresponding to the coupled Harniltonian given in ( 1.6). and where Hp sat-

isfies the spectral condition ( 1.3). ( 1.4).

The first result we present is the abstract PC estimate. Theorem 1.1.

This result is the basis for the spectral analysis of the Liouvillian. according

to the technique we outlined in the previous section. We point out that for

the PC estimate, we can allow an infra-red behaviour of the form factor as

g ( u ) -. 2. as Y + 0. with p > 0. which covers the physical case p = 112.

see (1.16).

In Theorem 1.2. we characterize the spectrum of the Liouvillian in view

of the property of Return to Equilibrium. .As mentioned in the previous

section. to prove this result. we combine the PC estimate with a Virial-

Theorem-type argument. It is for the latter that we need presently the more

restricting infra-red behaviour p > 2 (see (1.8)). We think that our method

can improved so that the physical case is included.

.As a corollary of Theorem 1.2. we show Return to Equilibrium in Corol-

lary 1.3 (see also Srction 2.1.4. Proposition 2.1).

-111 these results hold under assumption of the Fermi Golden Rule Con-

dition, (1.11). (1.12). In our last result (Theorem 1.4). we give an explicit

formulation of the Fermi Golden Rule Condition.

Theorem 1.1 (Positive Commutator Estimate). Assume the IR and

UV behaviour ( I . 8) , with p > 0. Let 1 be an interval containing ezactly one

eigenvalue e of Lo and let h E C,Z s.t. h = 1 on A and supph na(L,) = { e ) .

Assume the F e m i Golden Rule Condition (1.11) (or (1.12)) holh. Then.

for X small enough (independent of 3 ) . and mi fomly in 3 2 30 (for any

j b e d 0 < Jo < x). we have in the sense of quadratic fom on D ( L V ' / ~ ) :

Remarks. 1. We show in Chapter 2. Theorem 2.4 that L is essentially

selfadjoint on a dense domain in the positive temperature Hilbert space.

2. :\: is the number operator in the positive temperature Hilbert space (see

(2.28) and (2.29)). and Po,, is the projector onto the span of RJV0. the 3-

K M S state of the uncoupled system (see (2.23)). Xiso. is the Kronecker

symbol. equal to one if e = 0 and zero else.

3. The commutator [L. -41 is by construction in first approximation equal

to .V (see Section 3.1). and h(L) leaves the domain D ( W 2 ) invariant (see

Appendix A. Section As) . so that ( 1.17) is well defined.

Theorem 1.2 (Spectrum of L ) . Assume the IR condition p > 2 (see

(1.8)). T h e n the Liouvillian L has the following spectral properties:

1 ) Let E + 0 be n nonzero eigenvnlrre of L o , and suppore th.at the F e n n i

Golden Rule Condition ( I . I I J holds for e. Let .3 2 30 ( fo r any b e d

0 < :jO < m) , and X << 1 (independent of .3 2 &).

Then L has no eigenvalves in the open interval ( e - . t+ ). where E - i s

the biggest eigenval.ue of Lo smaller than E. and e+ is thz smallest

eigenvalue of Lo bigger than e.

2) Assume the Fermi Golden Rule Condit ion (1.12) hold . for e = 0. If X << niin{l. 3-'). then L has a simple eigenuahe at zero.

Remarks. 1. Theorem 1.2 shows that if the Fermi Golden Rule Con-

dition holds for all eigenvalues of Lo. then L has no eigenvalues. except a

simple one at zero.

2. Let Rj,x be the zero-eigenfunction of L. Then [BFS4] show that Rae,, is a

0-KMS state for the coupled dynamics. see also Section 2.2.2.

Corollary 1.3 (Return to Equilibrium). S-uppose the IR condition as in

Theorem 1.2, and that the F e n i Golden Rule Condition is s a w e d for all

eigenualues of Lo . T h e n every n o m a l state w.r . t . the 3-KMS state RJVA (the

zero eigenvector of L J exhibits return to equilibrium in a n ergodic meitn sense.

Remarks. 1. The Corollary follows immediately from Theorem 1 .2 and

Proposition 2.1. where the ergodic mean convergence is described in (2.8).

2. If one has more informat ion about the spectrum of L . then st longer modes

of convergence can be shown. For instance. absolute continuity of the spec-

trum (away from zero) shows pointwise convergence. see [BFS.I].

3. For the definition of normal states and IiMS states. we refer to Sections

2.1.2 and 2.1.4 respectively. We mention here that nornlal states w.r.t. a

IZvIS state are physically pictured as states that differ from the IiMS state

only in bounded regions of physical space ( asymptotic equilibrium states) .

-4s a nest result. we give an explicit formulation of the Fermi Golden

Rule Coadi tion. To do so. we need to introduce some notation. For a fixed

eigenvalue e # 0 of Lo. define the following subsets of N:

We also let, Pi denote the rank-one projector onto span{oi}. where HPoi =

Eioi and for any nonempty subset .V c N. set

Set Em - En = Em,. and for e E o( L,)\{O). rn E A$ and n E A;. define:

Here. the superscript denotes the complement. Xorice that if e = 0. then

,VT = i\i; are empty. and 5,. 6; = 0.

Theorem 1.4 (Spectrum of r ( e ) ) .

1 ) Let e # 0 and set & := inf,E,v*, {S,,) + infnE,\;; (8;). Then we have for

all 3 2 do (for any 0 < Jo < x; b e d ) :

In particular. the F e m i Golden Rule Condition (1.11) i s satisjied if the

r.h.s. is not zero.

2 ) Set for notational convenience r ( 0 ) = P ( L , = O)r(O)P(L, = 0 ) . T(0)

ha? an eigenvniue at zero, with the particle Gibbs state f?; as eigenuec-

tor:

,where 2&3) = tr e-'"p < x. Moreover. if

i strictly positive, then zero is a aJirnple ezgenvak~e of T ( 0 ) with ,unique

ezgenvector 0; and the spectrum of r ( 0 ) has a gap at zero: (0. 2goZp) <

20

a( F (0) ) . In particalar, the F e m i Golden Rde Condition (1.12) holds.

Remarks. 1. If E # 0 is finitely degenerate. then the infimum in the

definition of do is a minimum. If e is nonclegenerate. i.e. if e = Em,,, for a

unique pair (mo. no). then 60 reduces to

2. If H, is unbounded. then go = 0. Indeed. let m be fired. and take

n + m. then Em. < 0 and (0,. GO,) + 0. since On goes weakly to zero.

?jotice though that go > 0 is only a sufficient condition for the Fermi Golden

Rule Condition to hold at zero.

3. The size of the gap. 'go Z p is bounded away from zero uniformly in 3 2 .&. since

\vhere E, := Ei - Eo 2 0 (Eo is the smallest eigenvalue of H p ) and Hp :=

H,, - Eo 2 0 (the smallest e igendue of fi, is zero).

1.4 Review of previous results

Proving the return to equilibrium property is one of the key problems of non-

equilibrium statistical mechanics. Cntil recently. this property was proven

for specially designed abstract models (see [BRII] ) . The first result for realis-

tic systems came in the pioneering work of JakSiC and Fillet [JP1.21 in 1996.

In this work. JakSiC and Pillet prove return to equilibrium. with exponen-

tial rate of convergence in time, for spin systems (i.e. two-level s?*stems) for

sufficiently high temperatures. Their work introduces the spectral approach

to RTE. JakSiC and Pillet examine an two-level system coupled to a massless

bosonic field. whose Liouvillian they construct using the work of Araki and

Woods [AW] and Haag. Hugenholtz and Winnink [HHW]. The spectrum of

the Liouvillian. acting on a suitably transformed positive temperature Hilbert

space. the JaksliC-Pillet glued space. is analyzed in the spirit of Quantum Res-

onances. using spectral deformation techniques. and where the deformation is

generated by energy- t ranslat ion. The resolvent of the deformed Liouvillian.

for small values of the coupling const ant. has a meromorp hic continuat ion

from the upper half-plane into a strip below the real asis. with poles given

by the eigemdues of a so-called quasi-energy operator. The imaginary part

of those eigenvalues yield the reciprocal of the lifetimes of the resonances.

and they are of the order square of the coupling constant. A'. provided the

Fermi Golden Rule Condition is satisfied.

The IR condition on the form factors is g(&) -- d'. ; + 0. with p > - 112.

hence includes the physical case p = 112. However. the complex deformation

pushes the continuous spectrum of the Liouvillian below the real asis by a

distance only proportional to 113 (which is proportional to the temperature

T). This results in the condition X < 1 / J for the coupling constant. hence

the argument is not valid for small temperatures. Due to the method used.

certain analyticity conditions on the form factor are required.

The S-level system coupled to the massless bosonic field is treated in

[BFS.I], but the spectrum of the Liouvillian is analyzed using complex &la-

tion instead of translation. RTE with exponentially fast rate in convergence

in time is established for small coupling constant X i n d e p e n d e n t of 3. Bach.

Frohiich and Sigal adapt in this work their Renormalization Group method

developed in [BFS1.2.3] to the positive temperature case. The IR condition

is p > 0. which includes the physical case.

In a recent work. Dereziriski and JakSiC [DJ] consider the Liouvillian of

the :\*-level system interacting with the massless bosonic field. Their anal-

ysis of the spectrum of the Liouvillian is based on the L i m i t i n g A b s o r p t i o n

Principle and the Monrre Theory. which can he viewed as an infinitesimal

version of the spectral deformation theory. Dereziriski and .JakSiC choose for

the acljoint operator in the Mourre Theory the energy-translation generator.

as in [.1P1.2]. The IR condition for instability of nonzero eigenvalues is p > 0.

and for the lifting of the degeneracy of the zero eigenvalue. it is p > 1. which

does not include the physical case.

The basic idea of [D.J] is to show a strict global hlourre estimate away

from the tacuurn sector. By global. we mean that the Mourre estimate does

not hold only on compactly localized spectral subspaces w.r.t. the Liouvil-

lian (in other words. there is no EA( L ) in the Uourre estimate (1.1)). Then.

using this Mourre estimate. JakSiC and Pillet show the Limiting Absorption

Principle away from the vacuum sector. Using the Feshbach map, it is then

shown, under the assumption that the Fermi Golden Rule Condition holds,

that the Limiting Absorption Principle holds on certain intervals on the real

asis. showing absolute continuity of the spectrum of the Liouvillian there.

If the Fermi Golden Rule Condition does not hold (which is the case for

the zero eigenvalue). their result gives an upper bound on the number of

eigenvalues that may survive the perturbation.

The method for the spectral analysis of the Liouvillian we use employs the

energy-tranlation generator in the JakSiC-Pillet glued positive temperature

Hilbert space. as in (JP1.21 and [DJ]. However. unlike [DJ]. we do not aim

at a global Mourre estimate away from the vacuum sector. but rather at

a Mourre estimate that holds ou spectrally localized subspaces w.r. t. the

Liouvillian (hut without the restriction of being away from the vacuum).

This method has been developed in the zero-temperature case in [BFSS]

(for the dilation generator though) and it is based on a modification of the

bare translation generator. We believe that this met hod is simpler and more

robust than the one used in [DJ].

Our construction of the PC works for the IR condition p > 0. which

includes the physical case.

In order to conclude absence of eigentdues from the PC estimate, the

Virial Theorem is needed. It turns out that the systems for which the Virial

Theorem was applied so far have always satisfied the condition that [L. A]

is relatively bounded with respect to L. in which case a general theory has

been developed. see [ABG] (for specific systems. see also [BFSS] for particle-

field at zero temperature, [HSl.'Z] for -I+-body systems). It should be pointed

out that the Virid Theorem is an important tool of interest on its own. still

currently under research. see e.g. [GG].

In this work. we develop a Virial-Theorem type argument in our case

where the commutator [L. A] is not relatively L-bounded. But this comes at

the price that our result involves the triple commutator [[[L . -41. -41. -41. and

consequently, we need a restrictive IR behaviour of the form factor. namely

p > 2. We think that this restriction coming from the part of the proof using

the Virial Theorem (not the PC estimate). can be improved by a better

understanding of the Virial Theorem.

We finish this comparison of results by mentioning that in order to show

the lifting of the zero eigenmlue degeneracy. we need the condition X < 113.

1.5 Organization of the thesis

Chapter 2 is devoted to a mathematical introduction to quantum stat isti-

cal systems. We start by introducing general quantum statistical systems

described by a C'-algebra of obser\ahles and a dynamics on it. We then

explain how to represent the dynamics on a Hilbert space ( the GNS space)

via the Liouville operator. We introduce positive temperature systems and

equilibrium states (KMS states) and show how return to equilibrium is char-

acterized in terms of the spectrum of the Liouvillian. In Section 2.2. we

define the particle-field system at positive temperature. including a discus-

sion of the equilibrium states ( K M S states) of the uncoupled and coupled

system. One of the important steps in this chapter is the introduction of

the Liouvillian of the system in question. and showing its selfadjointness (see

Theorem 2.4).

Chapter 3 contains the proof of Theorem 1.1. In a first step. we use the

Feshbach method to show a PC estimate w.r.t. spectral localization in the

uncoupled Liouvillian Lo. Then. this estimate is estended to spectral local-

ization in the full Liouvillian L. showing Theorem 1 .l.

In Chapter 4. we prove Theorem 1.2 by developing a new Virial-Theorem

type argument. The proof of Theorem 1.2 uses the result of Theorem 1 .I. We

finish this chapter with a result on perturbation of K M S states. establishing

a link between the 1iMS states of the uncoupled and coupled system.

Appendix .-\ contains the proofs of Theorem 1.4 and some Propositions

used in the proof of Theorems 1.1 and 1.2.

In Appendix B, we give a brief overview of standard concepts in quantum

field theory used in this work. We also give an outline of an operator calculus

we use extensively.

The Bibliography is found at the end of the thesis.

Chapter 2

Mat hemat ical framework

2.1 Quantum statistical systems

In this section. we introduce the basic notions of quantum statistical systems.

in particular positive temperature systems. The most important object we

define is the Lio*uviZZian. which is the generator of the time translation.

For detailed information on quantum statistical physics. me refer to [BRI.II]

or also [S].

2.1.1 Observables and states

Consider a quantum system determined by a Hilbert space 'H and a Hamil-

tonian H on R. If the system at time zero is in the state c*. then according

to the Schrodinger evolution. its state at time t is given by a ( t ) = eeltH u*.

The average of an observable a in the state a * ( t ) is

Xext , assume we have only some statistical information about the initial

condition of the system: assume that with probability p,. the system is

initially in the state &, where En p,., = 1 and {on } is a family of normalized

elements in R. By this. we mean that the average of the observable a in the

state yT at time zero is given by

and it evolves accroding to

Relation (2.1) call be rewritten as

(a)a = tr pea. po = C P" Lo n

where po is called the density matrix of the system. and Po, is the rank-one

projector onto span{on). Clearly. the density matrix satisfies

We define the time evolution of the density matrix as

so that. taking into account (2.2) and the cyclicity of the trace:

By formal differentiation (notice that in general. H is unbounded). we obtain

the von-Xeumann equation

This discussion motivates a more abstract definition of obsertabies and

states of a quantum statistical system.

The observables are by definition elements of a Cm-algebra It. called the

algebra of obserlables. In the above discussion. 2l = B ( 2 ) . the C'-algebra of

bounded operators on 31. The evolution of observables is given by a strongly

continuous one parameter group of *-automorphisrn. at : II -i U. t E R. In

the above language. we have a t (a ) = ~ " ~ n e - " ~ . the Heisenberg evolution.

.A state over 2l is a normalized positive linear functional over the algebra

of obser~ables. i.e. ct : 2 + C linear and such that

In the situation above. ,. corresponds to n r tr pou.

2.1.2 Representations oft he algebra of observables and t he Liouvillian

Let (a,,.. a t ) he a quantum statistical system as introclucecl in the last

paragraph. To every pair ( P L ) . one can associate a cyclic representation

( T 0). i . . a Hilbert space R. a *-morphism 7 : 9 + B(R) and a nor-

malized vector R € 'If with the property that x (2 l )R is dense in 7f and

; ( a ) = ( R . a ( a ) R ) . Va E 3. (2.3)

The construction of (31.a.R) starting from (Zl .~. .n~) can be done in an

abstract setting (i.e. for an abstract Cm-algebra with a state i.). it is

called the GNS-con~truction (Gelfand-himark-Segal). see e.g. [HI or [BRII].

The triple ('H, a. R ) is called the GNS-representation of (a. J). The GXS-

representation is unique up to unitary equivalence. which means that if

('H. T . R ) and (HI. a'. 0') C-e two cyclic representations of (Xd) satisfying

(2.3). then there is a uniquely determined unitary map L' : H + R' such

that

If the state w is in\ariant under the evolution a t . i.e. if o at = c.. Vt . then

by the uniqueness of the G?iS-representation. there is a uniquely determined

one-parameter group of unitary operators [ ' ( t ) : '?l + 31 s.t. V n E a.

a ( a t ( a ) ) = ~ i ( t ) - ' a ( a ) l - ( t ) and L-(t)R = R. (2.4)

If t H i , ' ( t ) is strongly continuous. then the Lio~vi l l ian L is by definition the

selfacljoint generator of i t ( t ) (Stone's theorem):

?jotice that since CW(t)R = R. then L R = 0. i.e. R is a zero eigenvector of

the Liouvillian.

By construct ion. the Liouvillian generates the trivial dynamics of the

invariant state d . There is a natural "neighbourhood" of the invariant state

in the space of all states. for which the dynamics is also generated by the

Liourillian constructed from ;. This neighbourhood consists of the no7mal

states with respect to 7. the GXS-representation of d. .A state J' is called

normal m.r.t. a (or m.r.t. d) iff Va E 3:

n>_l

for some { p , ) c [O. I]. En p. = 1 and some {a?,) c R. 11 v,ll = 1 Vn. From

(2.4) and (2.3). it is clear that

s t ( a t ( a ) ) = Pn (C,. n(a t (a ) )d tn )

so the dynamics of w.' is indeed determined by L.

2.1.3 Positive temperature systems

So far we gave a general description of quantum statistical systems. Let us

now introduce positive temperature systems by defining special states that

depend on the parameter 3 = (ksT)- ' > 0. called the inverse temperature.

Here. ke is the Boltzmann constant and T is called the temperature. These

special states are first defined for finite systems and are called Gibbs states.

or epilibriwn states at temperature 3.

By definition. if the Hmiltonian H of a system satisfies tr r - j H < x;.

for 3 > 0. then the system is called finite. ?jotice that a finite system

has a Hamiltonian with purely discrete spectrum. otherwise the trace is not

defined. By analogy mi th classical statistical mechanics. the equilibrium state

at temperature 3 for a finite quantum statistical system is given by the Gibbs

state at temperature -3:

Using the cyclicity of the trace. we obtain

.A state satisfying (2.6) is called a $ - INS state. and relation (2.6) is called

the IiMS con& tion ( Iiubo-Martin-Schwinger ). and it follows from it that

i.e. the evolution at leaves the state LJJ invariant. This shows that every

($)-IiMS state is time translation invariant (stationary). Xotice that the

converse is not true.

For finite systems, the equilibrium state at temperature .3 is simply de-

fined to be the Cibbs state (which is a I<4IS state). For infinite systems. it

is not clear how to define equilibrium states. since a Gibbs state does not

exist in the sense that the Hamiltonian is not trace class (the spectrum of

the Hamiltonian describing an infinite system contains continuous parts).

Clearly. invariance under time evolution is a physically necessary require-

ment on an equilibrium state. but it should not be a sufficient one. Indeed.

a natural definition of an equilibrium state for an infinite system should con-

tain the aspect of at tractivity in a dynanlical sense (see the uest suthsection).

Defining the equilibrium states of an infinite system (at temperature 3 ) as

being the 3-KSIS states is then natural for two reasons: firstly. performing

the thermodynamic limit (see below) of a finite-volume Gihbs state preserves

the K M S condition. i.e. the limit state is again a IiMS state. Secondly. we

show in this work that the KMS state (of the coupled system) is dynamically

at tractive. i.e. its normal states eshibit return to equilibrium.

We now outline the procedure of the thermodynamic limit. In order

to define the analogous of a Gibbs state for an infinite system. one can

imagine that the physical system under consideration is first confined to a bos

.I c W3 of finite volume. Imposing boundary conditions on a:\ (Dirichlet or

von Xeumann), the Hamiltonian HA, governing the evolution of the confined

system is trace class. and consequently. one can then define the Gibbs state

of the confined system:

Here. the observxble a belongs to a local C'-algebra of observables 21f. Typ-

ical elements of this local algebra can be imagined to be functions of x E R3.

with support in .\.

The evolution on Qh is the Heisenberg evolution. i.e. it is given hy the

strongly continuous one parameter group of autornorphisrns t c, a;\. where

Xotice that 4; is a .3-Ii31S state w.r.t. . To obtain the infinte system.

one passes to the thermodgnamic limit. whcrrr the size of the confining box

tends to infinity through a sequence .\, t R3.

For a concrete model. one then proves (or for a general theory. one as-

sumes) that ~ ; \ ~ ( a ) converges as n + s. Va E %, := u,&. hence defining

the %mi t evolution"

for all n E as. ctt can then be extended by continuity to a *-morphism on

the C'-algebra 2l = Q,. where the bar means norm cIos.ure. This C'-algebra

is called the algebra of quasilocal obsenlables. and it is smaller than the C'-

algebra of all bounded operators (which could result as the weak closure of

3, ). In particular. bounded functions of the total energy operator H do not

belong to this quasilocal algebra. hence axe not observables for the infinite

system.

Finally, one shows (or assumes. in a general theory) that the numerical

sequence din ( a ) converges (for any a E 21,). defining a state

first on 21,. then on its closure Q. One shows that &?,J is a IiMS state with

respect to nt over the CW-algebra 2. We say the KSIS condition survives

the thermodynamic limit. Clearly now. dj is a natural candidate for the

analogous of the Gibbs state for an infinite system.

2.1.4 Return to equilibrium, spectral characterization

Let &J be a 3-IiMS state m.r.t. the dynamics a t . and let (R. r . 0). L be its

GYS-representation and the Liouvillian. respectively. as defined in Subsec-

tion 2.1.2.

h state J is said to exhibit return to eq.trilibriwn if Va E 9:

where the mode of convergence has to be specified. Physically. an equilib-

rium state should be locally attractive in a dynamical sense. i.e. ( 2 . 7 ) should

hold at least for Y.' in a -natural neighhourhood" of dj in the space of all

states.

We now give a criterion on the spectrum of L that guarantees return to

equilibrium for all normal states w.r.t. a.

Proposition 2.1 (Return to Equilibrium). Let dj be a 3-KMS state

with respect to the dynamics 01. and denote the GNS representation of 33 by

( X . T . OJ). Suppose that the Liouvillian L generating the dynamics on R has

n o ezgenvalues except for a simple one at zero. so that the only ezgenvector

of L is ~ 9 ~ . Then. for any normal state ," .cu.r.t. ii, and for arty obsemable

.-I E 3. we have

This means that the qs tem exhibits return to equilibrium in an ergodic mean

sense.

Proof. Suppose we can show (2.8) for L.' of the form

for any normalized cq E R. Then we get

where we used the Dominated Convergence Theorem (not ice that

and Cr=l pn = 1 in the last step. We conclude that in order to show (2.8)

for any normal state. it is enough to shotv it for d' of the form (2.9). which

is what we do now.

Since OJ is cyclic. then 9" := a(Bn)(P3 i a. in the norm topology of

'H, for some sequence {B,) C 3. It follows. using llir(ctt(-4))ll 5 IIAII. that

V€ > 0,

provided m. n 2 .V(c. A). urliformly in t . L k used the Schwarz inequality.

Therefore,

T = lirn 1 (Cn. ?(a,( -4) I$"') dt + O ( r )

T + x T

?;ow t.~sing the IiMS condition for d ~ . mc get. with 7r(nt(.4)) = ~ ' ~ ~ a ( . - l ) e - ' ~ ~ :

where we also used the fact that LQJ = 0. Since the only e igendue of L is

0. and it is simple. mi t h eigenvec tor &. then

tv- lirn - p L d t = p4,

where Pa, is the projector onto the span of (D3. This is a consequence of the

swcalled R;\GE theorem (see e.g. [CFKS]). Therefore. the limit in the A s .

where we used the I<MS condition in the last step and LcDJ = 0 in the step

before. Clearly.

which finishes the proof.

2.2 The particle-field system at positive temperature

In this section. me construct the Lionvillian governing the dynamics of the

system described by the Hamiltonian introducecl in Section 1.2 in a Hilbert

space setting (on the GSS Hilbert space).

2.2.1 The non-int eract ing system

Hamiltonian, Observables

We recall that the Hilbert space of the system (at zero temperature) is given

by (see Subsection 1.2.1)

and the non-interacting Hamiltonian is

where we omit now the trivial factors flf and I,. Here. the field Hamiltonian

was introduced in (1.5). It is a positive selfadjoint operator on its natural

domain

The particle Hamiltonian is a semibounded selfadjoint operator with dense

domain V( H p ) c 'H,, with purely discrete spectrum as given in ( 1.3). ( 1 A).

Clearly. KO is selfadjoint on the dense domain

The algebra of observable~ is given by the C'-algebra

where U(R) denotes the bounded operators on R. and W ( r j ) is the Weyl

CCR-algebra (see Appendix B for details). The t ime evolution on 3 is given

by the strongly continuous one-parameter unitary group of *-automorphisms:

Equilibrium state of the non-interacting system

The equilibrium state for the combined system is given by the product of the

equilibrium states of the single systems. namely

Here. 4 is the particle-Gibbs state at temperature J. and s: is a field I<MS

state at temperature 3 .

Let us consider first the particle-Gibbs state.

Since the particle system is a finite system. the construction of the GSS-

representation for (B(X,).d5) is standard. The GXS Hilbert space can be

realized as X(B('h!p)). the space of Hilbert-Schmidt operators on 'H,. and the

cyclic vector is given by e - ~ ' ~ p / * / J-. We identify *&( B(R,) ) 2 Up2'H,

and obtain the following representation:

J

where {o,} is the basis of R that diagonalizes the particle Hamiltonian.

HPo, = E,oJ. It is straight-forward to check that d5 is a 3-IiMS state for

the evolution

The construction of the (infinite) field KlIS state is more involved. and

it was done by Araki and LVoods [.iiV]. The Araki-Woods representation of

the Weyl CCR-algebra is given by the triple ('H;\\v. r-qiv. R.4u4. where

Recall that 3C is the Fock space over the one-particle space fj = L2(W3. d3k).

and R is the kacuurn in 8 ,. The function p = p(k) is the momentum density

distribution. given by Planck's law:

p ( k ) = 1

s = IPI. eJa - 1

which describes black body radiation. In terms of the creation and annihila-

tion operators. a;l\r is given by

The vector Rltrv is called the Araki- Woods vacarm. and it defines a vector

state over W ( 4 ) by

hraki and Woods show that II,\\,/ is a cyclic vector for n..t\v. so the Araki-

Woods representation is a GYS-representation of L:. It is not difficult to

show that d< is the unique '3-KblS state with respect to the evolution

The combined non-interacting system has a 3-IiMS state given by (2.12)

with respect to the evolution (2.11). and its GXS-representation is given by

(R. a. Od,O), where

f Since w$ and uJ are invariant under the evolutions (2.14) and (2.16) respec-

tively. then do.0 is invariant under the evolution (2.11). As discussed above.

(see (7.4). (2.5)). this guarantees the existence of a selfadjoint operator Lo.

the non-interacting Liouwillian. such that

It is not difficult to see that

Xotice that Lo is not semibounded. even though Hp aud HI are.

2.2.2 The interacting system

The interacting Hamiltonian

The total Hamiltonian was introduced in Subsection 1.2.1. see (1.6).

Theorem 2.2. The Hamiltonian H given in (1.6) i s selfadjoint o n the dense

domain D ( H ) = D ( H o ) = Z) (Hp) 13 D ( H f ) .

Proof. Xotice that r . is Hi-bounded with relative bound zero: indeed.

if cr# denotes creator or annihilator. one has (for the first inequality see e.g.

the proof of Proposition 2.3)

for any E > 0. Then the Kato-Rellich theorem yields the result.

Equilibrium state of the interacting system

Our nest task is to give the GXS construction for the interacting system.

and to find the Liouvillian that implements the dynamics determined by the

interacting Hamil tonian H.

Consider the Hilbert space R given by (2.17). We define the anti-linear

representation a' : 2l i B(R) hy

where C is the operator of comples conjugation in the basis of Rfl, that diag-

onalizes the particle Hamiltonian Hp. and

T ; ~ - ( I . C - ( f ) ) = I,V(&~) :; CFw(dmf).

In terms of creation and annihilation operators. the last equation reads

We now define the interacting Liouvillian as

where we defined

In the nest paragraph. we show selfadjointness of ( a oni tardy transformed

version of) the Liouvillian. Let us now esplain some properties of L that

justify its definition as the Liouvillian of the interacting system. For details

and proofs, we refer to [BFS-L].

Define the selfadjoint operator

then one shows that fiJ,o given in (2.18) is in the domain of LI. and that

defines a 3-KSIS state

for the coupled evolution

X a c t a , ( a ) = e i t Hae-itH

The dynamics is unitarily implemented by C. i.e.

a ( a : ( a ) ) = e i tL i i ( a )e - " ' . and f OJJ = 0.

so L: is indeed the Liouvillian of the coupled system. and the state d ~ , ~ is the

equilibrium state of the coupled system at temperature ,3.

2.2.3 JakgK-Pillet gluing, unitarily transformed Liou- villian, selfadjoint ness

Let us denote by 3(5) the Fock space over a space S. In view of finding

a positive commutator. it will be more convenient for us to consider the

unitarily transformed Liouvillian

where U acts trivially on the particle space. and

The unitary map V2 : 3 ( L 1 ( R 3 ) -+ L2(R3)) i 3 ( L 2 ( W x SZ)) is given hy the

lifting

where

We also define V& = R. where the Rs denote the wcua of the corresponding

spaces. We call V2 the Jakizi- f i l le t gluing. and 3( L2(R x SZ) ) we call the

JaliSiC-Pillet g h e d space.

The unitary map Vl : 3(L2(W3)) 3 3(L2(W3)) + 3(L2(R3) f LZ(R3)) is

determined by

and again, we define Vl (R 13 R ) = 0. the Rs being the Lacua of the corre-

sponding spaces.

We mention that under this transformation. the I<MS state (2.18) of the

uncoupled system is given by

where we recall that RPJ is given in (2.13) and R is the vacuum in .F( LL(R x

s2 ) ) In order to calculate ULIU- ' . we use the following relations. which are

easily verified:

v l e i t d T ( & ) :3 , - t t c I l ' ( ~ ) v r l = Vlr'(e't") 8 ~ ( e - ' ~ ~ ) v ; '

- - r ( p + e-y.

where we recall that u E R is the first variable of functions in L2(R x S2).

Differentiating with respect to t at t = 0 yields then

To calculate I = U I U - l . me use the following readily obtained formulas

( toget her with their adjoints)

where \>o - = - t r ) is the indicator function of ir E (0. x). and \<o is the

indicator function of u E ( -;. 0). One obtains:

where we recall that the arguments of functions in L2(W x S2) are called

( u t a ) E R x S'. We haveset for g E L2(W+ x S'):

and

Here, y ( u . a ) is the representation of g in spherical coordinates ( u . a ) E

P+ x SZ. Xotice that g l ( t l . a ) = -y2(-t1.n).

This shows that

L = Lo + XI,

where I is given by (2.24). and Lu = L,+ L,. mith L, = C p and Lj = dr(cc) .

Our next task is to show selfadjointness of the Liottvillian L acting on the

positive temperature Hilhert space

'N = 'R, :3 R, :3 3 ( L 2 ( R x S ' ) ) .

To do so. we start off by introducing the positive operator

with domain D(.\) = {i E 'H : Il.\li!ll < x.}. We also set

Proposition 2.3 (Relative Bounds). Set L2 = L2(R x S 2 ) , and let 0 < < m be a jixed number.

3) For ut E V(.V'/') and C? E D(.\'/') respectively, .we have the following

bounds. uniformly in 3 > ,Jo :

Here. C < C'(1 + 3;') . where C' is independent of 3 . Jo.

4 ) For d* E v(s'/~). any c > 0. and uniformly i n 3 2 &. me have

5 ) For c7 E V(.\'12). any c > 0 , and uniformly in 3 2 Jo. we have

Remark. The parameter 30 gives the highest temperature. To = l/.jO.

s.t. our estimates 3)-5) are d i d uniformly in T 5 To. Notice that To can be

fixed at any arbitrary large value. Since we are not interested in the large

temperature limit T + x;. we set from now on for notational convenience

To = 1.

Sotice that 4 and 5 ) tell us that Qc > 0.

where we understand these inequalities holding in a sense of quadratic forms

on D(!v'/*) and v(:V/~) respectively.

Proof of Proposition 2.3. 1) Csing the Schwarz inequality. we have Vv:

a - d , 4 (1 lj( u , a), u . a)*v-%,l)2

5 11 f 1 (.v-l4!*. c l = ( u . a)a(u. * ) * V - ~ ~ C )

= /If Il& l1412.

2) Proceeding as in 1). we get

and using the CCR [ ~ ' ( f ) . a ( ~ ) ] = (f .g) . we get

II~Ygr.2)~?11~ = (G a(91,2)a=(g1,2)~7) = I l~(gi ,2)~II2 + Ilgl.:!il$ll t:I12-

SO

2 ) above, we get

Xext, we show that l/glIIL2 5 C and 11 l ~ 1 - ~ / ~ ~ ~ 1 1 ~ z 5 C. uniformly in 3 2 4,. Indeed. notice that

/lgl11:2 = L3( l + ?p)lg(h?. a)2c~ulidS(n) = llg2 I:,.

where me represented g in the integral in spherical coordinates. Since we

3 2 3,. we get with (1.8) (for p > 0) the following uniform bound in .3 3 .30:

[ l g l , . 2 1 1 ~ 2 5 2 L 3 ( l + .l;1d-1)lg(k)12d3k = C < x. ('2.30)

Similarly.

uniformly in ,L? 2 3,. It is clear from the last two estimates that C satisfies

the bound indicated in the proposition.

4) For any c. E 2)(W2) and any c > 0:

The result then follows from (2.30).

5 ) is proved as 4).

These relative bounds and Selson's commutator theorem yield essential

selfadjointness of the Liouvillian:

Theorem 2.4 (Selfadjointness of t he Liouvillian). Since H, is bounded

below, there is a C > O s.t. H p > -C. Suppose that [G. H p ] ( H p + C)- ' I2 is

bounded in the sense that the quadratic f o r m c1 ct 2iIm (G'a.. Hpc*). defined

on D( H,), is represented by an operator denoted [G. H,],, s. t. [G. Hp],( Hp + C)-'/' is bounded. Then VX E R. L is essentially selfadjoint o n

Proof. The proof uses ?;elson's commutator theorem (see [RS] . Theorem

S.37). Let

then .V is selfadjoint on Do and .V 2 1. Also. L is defined and symmetric

on Do.

According to Yelson's commutator theorem. in order to prove Theorem

1.2. we have to show that V v E Do and some constant d > 0.

Estimate (2.31) easily follow from I I L P W 1 I I 5 1. I I L.jN-' ( 1 5 1 and / I I;\*-' 1 1 5

I l f (:I + ~)-'''II ll(i1 + 1)1/2(.\ + l)-'II 5 d (by 3) of Proposition 2.3).

To show (2.32). notice that Lo commutes with A'. so the 1.h.s. of (2.32)

reduces to

where

Let us esamine the first term on the r.h.s. of (2.33). It is easily shown that

since I ~ l g ~ , ~ E L2(R x S'). then am(gl,2).1 = .\n'(gl,2) + cf(ltr(gl.z) on D(.\). This shows that leave V(.\) imariant and so w e have Vc. E Do:

where me used Proposition 1.1 in the third step.

Yotv we look at I< given in (2.34). Using the specific form of I (see

(2.24)). we can write

where

We examine Kl. Let c* E 130. then ( H p + c)'/'(L* E R. a d SO

i = 2iIm (GI :: (n(g,) + a9(gl )). ( Hp + C) :3 It-) = ZIm ((~(gl) + am(gE ))L*. [G. H,],c*) .

so we obtain

The same estimate is obtained for ILl in a similar way. This shows (2.32)

and completes the proof. I

Chapter 3

Proof of Theorem 1.1

In this chapter. we give the proof of Theorem 1 .l. It consists of two main

steps: the Positive Commutator estimate w.r.t. spectral localization in the

uncoupled Liouvillian Lo (Section 3.1) and a passage from this estimate to

the one localizecl m.r.t. the full Liouvillian L (Section 3.2).

Our estimates are uniform in .? 2 Jo (for any 0 < Jo < x fixed). but as

we have mentioned in the remark after Proposition 2.3. we set .;lo = 1.

3.1 Positive commutator with respect to spectral localization in Lo

We construct an operator B (see (3.6)) which is positive on spectral sub-

spaces of Lo. The main result of this section is Theorem 3.4. which shows

positivity under assumption of the Fermi Golden Rule Condition.

On L2(R x S2) and for t E R. we define the unitary transformation

( i ; r i * ) ( u . a) = B . ( U - -t. a ) .

which induces a unitary transformation LYt on Fock space 7 = 3 ( L 2 ( W x S2 ) ) :

i.e. for ti* E 3. the projection onto the TI-sector of L; el is given by

( & c l ) , ( t t l . . . . . a,,) = i * , ( u l - t.. . . . tr, - t ) .

Here and often in the future. we do not display the angular variables a\. . . . . a,

in the argument of L*.. L; is a strongly continuous unitary one-parameter

( t E R ) group on 3. Its anti-selfacljoint generator .-lo. defined in the strong

sense by at = .-lo. is

The domain of the t~nbotinded operator .Ao. V ( & ) = {c* E 3 : lt,oL,-t e. E

7). is dense in 7. which simply follotvs from the fact that .-lo is the generator

of a strongly continuous group. From now on. we write

Proposition 3.1 On the derwe set V(Lo) n V ( X ) . we have

where It b obtained from I by replacir~g the f o m factor g by ib traraslate g' .

and g t ( u . a ) = g(u + t . a ) . We obtain therefore

where i = GI (3 ( a R ( & g l ) + a ( & g l ) ) - G, 3 (a-(bug2) + a(&gz)) . The d e ~ v a -

tive in (3. I ) is understood in the strong topology.

54

We give the proof in Appendix -4, Section A.2. On a formal level. we

have

which suggests the definition of the unbounded operator [ L . =lo] with domain

V([L. =lo]) = D ( S ! as

We point out that the operator [ L A o ] is defined as the r.h.s. of (3.2). and

not as a commutator in the sense of L A o - &L. On certain dense subsets

of its domain. [L. .-Iu] behaves like an ordinary commutator. as shows the

following proposition (see Appendis -4. Section A.3 for its proof):

Proposition 3.2 T h e se t Do n D(.V) n V(.-1,) is dense and for a n y I* in

this set, we have ( v . [L . .40]c.) = 2Re ( L I L .-lev). Recall that Do defined

in Theorem 2.4.

Remark that [L. .-lo] is positive on V( .V) n Ran Pk . where R is the vacuum

in 3. Indeed. from Proposition 2.3. it follows (take e.g. c = 1/41

so that

On the other hand. Pn[L. .Aa] Pn = 0. so if we want to find an operator that

is positive also on a. then we need to modify .-lo.

For a fixed eigenvalue e E a( L,). define

Here. 6 and e are positive parameters. and Q. are projection operators on

'H defined as

- Q=P(L ,=+:2Pn . Q = n - Q .

In what follows. we denote

- = QR..

Proposition 3.3 The operator b = b ( ~ ) is bounded and [ L . b] = Lb - bL

is .well defined on and it extends to a bounded operator o n the .whole space.

We denote the extended operator again by [ L , b] .

Proof. The operator b is bounded since both I Q and Q I are bounded.

Furthermore. since llLoRcll 1 + lellc and llLoQll = lei. then [Lo .b ] is

bounded. Moreover. since llIQll 5 C and ~IIE;IQ~~ < Ce-'ll(-V+ 1)IQII 5

2Cc-'IIIQII 5 C E - ~ . then also II[I. b]ll < s. We used the fact that Ran I Q c Ran P( X 5 1). since I is linear in creators and .VQ = 0. I

We define the operator [L. A] by V([L. A]) = V ( 3 ) and

and the operator B by D ( B ) = D(.V) and

Were is the main result of this section:

Theorem 3.4 Let e E o(L , ) and let 1 be an internal around e not con-

taining any other eigenvalue of L,. Let EL be the (sharp) indicator function

of 1 and set Ez = E A ( L o ) . Assume that the Fermi Golden Rule Condi-

t ion ( I . 11) (or ( I . 12)) holds, and that the parameters satisfiJ 0. E << 1 and

e << 8. 8A2 << e3. Then we have on D(.\~~/'). in the sense of quadratic forms:

where Pn , , is the projector onto the span of RJ,o defined in (2 .23) .

The proof of Theorem 3.4 uses the Feshbaclr method. which w e explain in

the nest sect ion.

3.1.1 The Feshbach method

An essential ingredient of the proof of Theorem 3.4 is the Feshbach method.

The main idea of this method is to use an isospectral correspondence between

operators acting on a Hilbert space and operators acting on some subspace.

We explain this method adapted to our case. For a more general exposition.

see e.g. [BFS'Z].

Consider the Hilbert spaces 3, defined by

where \, = \ ( N 5 v ) is a cutoff in S. and Y is a positive integer. With our

definitions of Q. Q. (see (3.4)) we have

, - Define Q1 = \,E;Q and Q2 = \ , E ~ Q and set B, = QJ3QJ. 1.1 = 1.2.

The operators BiJ are bounded due to the cutoff in .V. Yotice that Q1,? are

projection operators (i-e. Q:,, = Q1,?) since 1, commutes with E i and Q.

By saying that : E @ is in the resolvent set of B2?. we mean that

( B22 - :)-I 1 Ran Qz esists as a bounded operator. The main ingredient

of the Feshbach method it the following observation:

Proposition 3.5 (isospectrality of the Feshbach map). If z b in

the resolvent set of B22 and if

then we have 2 E o#(B) : E a#(E) , ,where the Feshbach map El =

EJB) is d e f i l e d by

and a# stands for a or a, (spectrum or pure point spectrum).

The proof of Proposition 3.5 is given in a more general setting e.g. in

[BFS']. we do not repeat it here. We use the isospectrality of the Feshbach

map to show positivity of B in the following way (see also [BFSS]):

Corollary 3.6 Let do = info( B 1 Re) and suppose that B?* 2 JQ? for

some 4 > -m. and that inf a(&) 2 So u n l f o n l y in J for J 5 3 1 , wh,ere Yo

and d l are two f i e d (finite) numbers. Then we have d o 2 min{& inf a(&, )}.

Remarks. 1) All our estimates in this section will be independent of the

iV-cutoff introduced in (3.8). In particular. j . J o , JI,So are independent of

u. This will allow us to obtain inequality (3.7) on ~ ( : \ i l / ~ ) from the corre-

sponding estimate on Ran \: (LY u) by letting v + x (see (3.31 ) below).

2) The condition info(&) 2 Eo uniformly in J for J 5 d l . implies that

do # -x . Indeed. if do = -x;. then we have a sequence i), + -x.

3. E o ( B r H e ) . so by Proposition 3.5 (we show below that (3.9) is satisfied

for z = d,,. V n ) . we would have inf a(&,) + -x as n -t x. which is in

contradiction with the lower bound condition inf a(&) 2 20.

Proof of Corollnry .3.6. If do < J. then do is in the resolvent set of B22.

For : = do. (3.9) hoids. Indeed.

where we used the esplicit form of [L. b] and the fact that Q2Q = 0. With

where we used Ran IE: IP* C Ran P ( Y 5 2). and with

In a similar way, one obtains l lQIBQZ(B22 - JO)-'Q211 < X. SO (3.9) is sat-

isfied for z = do.

Now since do = inf o ( B I He) and do < j. there is a sequence J , + do.

J , < j. J , E a ( B [Re). and such that Vn. (3.9) is satisfiedfor 2 = d, (just

repeat the above argument and replace Jo by d,).

By the above remark 2). we have do > -CE, and therefore d o E a( B I Re) .

because the spectrum is a closed set. By Proposition 3.5. do E a(Ed, ). which

finishes the proof. I

3.1.2 Proof of Theorem 3.4.

The idea is to apply Corollary 3.6. to the operator

where is the Kronecker symbol. i.e. it is one if e = 0 and zero else. The

positive number 6 will be chosen appropriately below. see (3.29).

First. we show that Bi, 2 (311 - 6,,0 6)Qz (see (3.12)). then we shorn that

Ed 2 -1 - JeV0 6 =: So (Proposition 3 .7 ) . uniformly in J for 3 5 112 - (Iev0 5.

Invoking Corollary 3.6 will then yield the result. Notice that due to the

cutoff 1, in (3.S). Bi,. i. j E (1.2) are bounded operators. .Ill the following

estimates are independent of v.

- We first calculate Bi, = Q2B1Q2 Using QQ2 = 0. and de,oPkd,oQ2 -

&,-,Q2. we obtain from (3.10) and (3.6)

Proceeding as in the proof of Proposition 2.3. one shows that Vc > 0.

With our assumptions on g. 118.q1/l$ < x. uniformly in 3 2 1 (see (A.11)).

Using the inequality above with e = 1/10 and I ~ ~ I Q I I I 5 Ce? we obtain

As can be easily checked. Q2 = Q2Pk. so we have SQ2 3 Q2. and me conclude that

provided A' << 1 and 0X2 << c'. In the language of Corollary 3.6. this means

we can take J = 31.1 - SeQoS.

In a nest step. we calculate a lower bound on EJ for J 5 112 - S,,oS.

Proposition 3.7 We have. ~mifoonlg in J for J < l / 2 - ~5.,~d:

where the error t e rm is independent of 6 . Recall that RPJ is the particle Gibbs

state defined in (2.13).

Proof of Proposition 3.7 B y definition. Ed = B; , - B;, ( B;, - J ) - ' B;, . We show that B;, is positive and B;,( B;, - J)-I B;, is small compared to

BL-

With Q Q ~ = 0. QQl = Ql and J,,of&OQl = d,,oP&Ql. me obtain from 3

(3.10) and (3.6):

B;, = Ql (%X + + YBA'I~I - 5..6~&) Ql

- @ A 2 ) . (3.14)

where we also used ~i 2 - kX - 0 ( X 2 ) and QIS = 0.

Let us now esamine B;,(B;, - J)-'B;,. Notice that from (3.11). we get

where me defined the bounded selfadjoint operator Iil acting on Ran Q? as

[il = -p/? lo ( , \ ~ + 8 , \ 2 ( ~ ~ ~ ~ ~ - ~ ~ ~ ~ ~ ) ) . ~ - 1 / 2 . (3.16) 9

Therefore we can rewrite (3.16) as

where

where I< = (1 + IG)-'/~ is bounded and selfadjoint with I 11<112 = 111i211 =

c2. Wehavetherefore.from(3.19)and(3.17).and I K l + IG)-'Il 5 uniformly in J for il 5 112 - heVo6:

Notice that Bi2 = Bl? and B:, = B2'. Yaw. remembering ( 3 . 6 ) . and since

-VQl = 0 and Q2Q = O = B Q ~ .

thus with (3.20). we obtain

and so. together with (3.14). we get. uniformly in J for J 5 112 - 6e,oS:

We point out that the error term in the last inequality does not depend on 5.

With the choice of parameters we mill make (see (3 .50)) . (3.21) shows that

Ed 2 -1 - deqod uniformly in i) for J 5 112 - 6,,06. i.e. in the language of

Corollary 3.6. Xo = -1 - d,,06.

The remaining part of the proof consists in relating the strict positivity

of the nonnegative operator Q 1 IE: IQ to the Fenni Golden Rule Condition.

We let I, and I, = I,' denote the parts of I containing annihilators and

creators only. so that I = Ia + I,. Thtls

In the first step. me used I'Q I = 0 and Q I I, = 0 ( since I. Po = 0 ) and in the

second step. w used Q I,Q = Ql I. (since I,@ = 0). Sow write

where n2 is defined (1.10). and where we display the dependence of R: on e.

The operator-valued distributions ( a and a ' ) satisfy the canonical comrnuta-

t ion relations

[a(& a). aw(u' . d)] = 6 (u - t r t )&2 - a'). (3.24)

Nest. we notice that the pull-through formula

implies

a ( u . ct)RT(e) = Ri(e - u ) a ( u . a). (3.25)

Using the CCR (3.24) and (3.25) together with the fact that a ( u . a)QI = 0.

we commute a ( u . a ) in (3.23) to the right and arrive at

We can pull a factor Pn out of QI and place it inside the integral nest to

R f ( r - u ) and thus replace R:(e - u ) by ( ( L , - e + u ) ~ + e2)-l . Notice that

e ( ( L , - e + u)' + €')-I + S ( L p - e + (I) as E -+ 0. More precisely. we have

Proposition 3.8 For E << 1. w e have

Proposition 3.S. which we prove in Appendis .A. Section A.4. together with

(3.21)-(3.23) and (3.26) yields

which is (3.13). so Proposition 3.7 is proved.

Now we finish the proof of Theorem 3.4. If the Fermi Golden Rule Con-

dition (1.11) holds. then for r f 0. wc have T ( e ) 2 -,. > 0 on RanQl. so

we obtain from (3.13). and under the conditions on the parameters stated in

Theorem 3.4: Ed 2 ;i so by Corollary 3.6:

OX' inf a ( B I Re) > min{l/2. x d ~ ~ e - l - , , } = x-- i e . (3.27)

E

since by our choice of the parameters (see (3.50)). we have $ << 1.

For e = 0. me have r(0) = I'(0)Ph. since I'(O)RPJ = 0 (see Theorem 1.4). 3

so Proposition 3.7 gives

For some fixed 0

which gives with

(independent of 8. A. E) . we choose

Rememberiug that B' = B - dP&,O. we obtain from Corollary 3.6

ex2 inf a ( ( B - SP&J / ?-lo) 2 rnin(ll2. -2;ia0h2/e) = -2aa-.

E

from which we conclude that

n-here we used a l l o 5 -.

Estimates (3.27) and (3.30) yield Vc*:

8X2 (li.. \ & B E ~ \ ~ c O 2 -7. ( c ' . \YE;(1 - ~ 6 e . 0 P n d , - , ) E ~ \ w ~ ~ ) . (3.31)

E

Suppose now iy E V( .V1/'). Then. since ( .V + 1 )- ' I2 B( S + 1 is bounded

(see the definition of B. (3.6)). and since 1, + 1 strongly as v + nc. we

conclude that V v E D(.V'~'):

ex2 ( i v . E I B E O , ~ ~ ) 2 -ye (s.. Ei(1 - ~ S , , o P n , , ) E ~ ~ ~ ) .

E

mhich proves Theorem 3.4.

3.2 Positive Commutator with respect to spectral localization in L

We pass from the positive commutator estimate w.r. t. Lo given in Theorem

3.4 to one w.r.t. the full Liouvillian L. hence proving Theorem 1.1. The

essential ingredient of this procedure is the 111s localization formula. which

we apply to a partition of unity w.r.t. .b-. Then. me carry out the estimates

on each piece of the partition separately. Let

1 = ( x ) + ( x ) . .C E R+.

ii E CF([O. I]). he a Cr-partition of unity. For some scaling parameter

a >> 1. define

The reason why we introduce the partition of unity is that I\ I = O(al'') is

bouncled. Since the i leave D( .V1l2 ) invariant. then

[ i t . [ l a . B]] = \:B - 2\iB\, + B\?

is well defined on V ( . V 2 ) in the sense of quadratic forms. and by summing

over i = 1.2. we get the so-called IMS localization formula (see also [CFKS] ):

BII (3.32)

Furthermore. we obtain from (3.32) and (3.6). in the sense of quadratic forms

on v(;v'/'):

The proof of Theorem 1.1 is now carried out in four parts:

1. in Proposition 3.9. we estimate h ( L ) x 2 B k 2 h ( L ) .

2. in Proposition 3.10. we estimate h ( L ) k 1 B x l h ( L ) + +h(L)iVh(L) .

3. in Proposition 3.11, we estimate $ C,,? h(L)[\,. [ l i7 B]]h(L) .

4. to complete the proof. we choose relations among the parameters 0. A. E . a

(see (3.50)).

Proposition 3.9 i f X20 << €'a and A? << a. then w e have

Proof. Recall that B = 6.V + Af + [L. 61. Since Q = 0. we have Ye*:

Furthermore. Proposition 1.1 gives Vc > 0. X i 2 c 3 - 0 ( X 2 / c ) . so

where we picked the value c = 1/10. and used l 2 S r 2 2 61;.

- \ . IF; :XIF~, \ I + adjoint 2 -4

Taking c < 1/40 gives then

1 C-

-hXh 20 + h \ l F & ~ ~ ~ ~ , l i ~ + adjoint

- Nest. using Q F i t = 0 and ( L o - E ) Q = 0. we calculate

where we used I I R , ~ I I 5 lAfl-' 5 C and IIIklII 5 Call?. Sest . since -

supph n s u p p q = 0. then \IF!$h(L) = \ l K ( h ( ~ ) - h ( L & so by using

the operator calculus introduced in Appendix B. we obtain

P

h ) = J d ( ( ~ - ) ( L - ( L ) = O ( (3.41)

iFrom (3.10). we then have

which. together with (3.39) and (3.38) yield

- Our next step is estimating (3.3;). Again. using Q F:, = 0. we get

where we used ~i 2 -c.V - 0 ( X 2 / c ) and I IZR;~~ 5 I l ' I - ' 5 C. We thus

obtain. since 0 << I:

Finally. me investigate the positive term (3.35). By sandwiching ( 3 . 7 ) in

Theorem 3.4 (with E i replaced by E i , ) with F i t . and noticing that Fi, E i , =

Fx,. we arrive at

where me used (3.41) in the last step once again. and -?I:( F&)2Pn,, 2 -2 Pn,,, in the second step. Combining (3.45) with (3.42) and (3.43) yields

Proposition 3.10.

Proposition 3.11 We have

Proof. Sotice that 1, and 1 - \2 have compact supports contained in

[O. 21. ?;ow in the double commutator. we can replace t2 by 1 - 1 2 without

changing its d u e . So if suffices to estimate [ k (Sla). [k (Slla). B]]. where

E C,"([O.el). W e have

[\(.l'/o). [~ ( .V /O) . B]] = [\(.V/a). [!(X/a). ~i + [ L . b ] ] ] .

Notice that we have as operators on D(.v~/~):

and therefore, as operators on V( .v"' ):

[[urn( f). 31. -\.;I = a'( f).

Taking the adjoint yields on V( .W2)

and we conclude that on D(.\i5/')

and since (.Y/a - z)-l(.V/o - <)-'V(.Y~I?) = V(.'VS12). w e ha\-e in the sense

of operators on D(.v'/'):

We used the operator calculus introduced in the Appendix. Xow since

llf(N/o- :)-'/*[I 5 CII(:V+ l)1/2(:~/a-z)-111 5 C O ~ ~ ~ ~ I ~ ~ ~ - ~ . which follows

from

we conclude that

Yext , write for s iimplicity \ instead of \ ( .V/a). ancl look at [I. [ \ . [ L . b]]] =

@ A [ \ . [\. [L.E~ I Q ] ] ] + adjoint. Now [L.R: IQ] = S ~ , ? [ L ~ . I ] Q + X[I.E: I Q ] and

since (Lo - E ) Q = 0. From I @ L ~ - e ) l l 5 E-l and j l [ \ . (which follows easily from (3.46)). we have

Xext . consider

Use now

to arrive at the estimate

This together with (3.4;) and (3.43) yields

which proves Proposition 3.11. I

Sow we finish the proof of Theorem 1.1. The 111s localization formula

(3.33) together with Propositions 3.9-3.11 yield

0A2 0 5 @ A 2 h[L. =L]h 2 -7. (1 - O ( h L / ' ) ) h\;h + ?h\:h - 5--,oSc,ohPn,, h

€ n n E

The sum of the first two terms on the r.h.s. is bounded below by

so we get

Finally. we choose our parameters. Let e = a = x - " / ~ O O . 0 = hi/'00.

and choose

It is then easily verified that for small A. the concLitions on the parameters

given in Theorem 3.4 and Proposition 3.9 hold. and furthermore. (3.49) be-

comes

Chapter 4

Proof of Theorem 1.2

The proof is by contradiction using the following idea: assume c* is an eigen-

function of L with eigenvaluc e in A. 0 6 A. Then on one hand. we formally

have

and on the other hand. from the positive commutator estimate (Theorem

1.1). ([L. .-I0]),. should not be too small, which will give a contradiction. Since

we do not know whether c? is in the (form-)domain of the commutator [L. ;lo].

we replace it by an approximate vector c*,,, which is in the form domain

of the commutator. Here. 0.v stand for cutoffs in .-Lo and S respectively.

Then the PC estimate shows that ([L. 2 811 ~,,,11~. where 9 > 0 is

independent of the cutoff parameters a. u. On the other hand. we show that

([L. --Io]),3.v + 0 as a. u + 0. which yields a contradiction. The upper bound

on ([L. =lo]) tLa.u we call the Virial-Theorem type estimate.

For the zero eigenvalue. the argument goes similarly: assume zero is a

degenerate eigenvalue of L (we know that its multiplicity is at least one.

since we have a zero eigenvector RJ,x of L) . Then take an eigenvector a* E

Ran P&,A s. t . L uy = 0. .As above. the Virial-Theorem-type agument shows

that ([L. =lo]),a,u is very small. while the PC estimate from Theorem 1.1

gives a lower bound on this average value. Remark that here. we need to use

that the product Pn,, Pkd.A is small. which is satisfied provided 3X < C. see

Section 4.2.

4.1 The proof of Theorem 1.2 in the spirit of the Virial Theorem

We now prove Theorem 1.2 by implementing the idea given above. ;\ssume

ct is a normalized eigenvector of L with eigenvalue e. If E = 0. we assume in

addition that C y E Ran P&*, . Let cr > 0 and set

where f is a bounded E C" function. such that the tkrivatiw f' is positive

and s.t. f ' (0) = 1 (take e.g. f = Arctan). Set

f: := f t ( ia&). and h, := a. Furthermore. set fz := ft'(ia..lo). For v > 0 and g E C T ( 4 . l ) . define

v = g ( Here. a. v will be chosen small in an appropriate way. The

regularized eigenfunction is taken to he e*,,, = h , t8,. Notice that

Furthermore. we set for not at ional convenience in this subsection

Ii := [L. .-lo] = .V + hi.

W e now prove the above mentioned upper bound.

In the following. n.e often need to commute L through smooth functions

of :V and --I0. For this. me use Proposition A.2 in Section A.6.

From 2.. Proposition A.2. we find that

We used that ( L - E ) ~ J = 0 and that [X. I ] is .Y'12-bounded. Yext. observe

that

In the last step. we used

where we used the fact that .-lo and 3 commute in the second step. and (B.4)

with p = 1 in the last step.

.According to Proposition A.3 (Section A.6). we have in the sense of op-

erators on 2)(X'I2).

1 [ I . i fa] = f: [ I . .-lo] - ;So f t a d i o ( I ) + R.

where adto ( I) is the Mold commutator (agqo ( I) = [[ I . =lo]. .-Lo] ) . and where

(R) , " = 0(c22u-''2). Xotice that it is here that we need Iladio(l)L~1/211 5 C.

k = 2,3. hence the more restrictive IR behaviosr p > 2. We obtain from

(4.4) and recalling that i = [ I . ;lo]:

1 (4.3) = Re (fi K ) , u - 3 ~ e (icr f,"adio ( I ) ) + ~ ( a ' v - ' / ~ - C u 1

We used in the last step that

since ac4\0(I) is .Y'/'-hounded. Conlbining (4.5) and (4.2). we obtain

Next. we rvant to establish a lower bound on the last 1.h.s. by using the

positive commutator estimate. Let A be an interval containing esactly one

eigenvalue. e. of L,.

We introduce two partitions of unity. The first one is given by

where ka E Cm(-\). t A ( e ) = 1. We localize in L. i.e. we set k~ = k A ( L ) .

The second partition of unity is given by

where E CZ is a -*smooth Heaviside functiono7. i.e. t(s) = 0 if x 5 0 and

( s ) = 1 if x 1. We set for n > 0: tn = ( n ) . = 1 - W e will

choose n < l / v . so that

The last equation will be used freely in what follows.

We are going to use the INS localization formula (3.32) with respect to

both partitions of unity. and we start with the one localizing in .V:

where we used that I< 2 n / 2 on Ran P& and the estimate (3.17).

Yext. we investigate the first term on the r.h.s. From the 111s localization

formula for the partition of unity w.r.t. L. we haw

and we have used in the second step the fact that

LVe recall that b is a bounded operator (see Proposition 3.3).

In the last step in (4.5). we used the positive commutator estimate. The-

orem 1.1. in the following way. For e # 0. Theorem 1.1. gives right away

where me recall that [L. A] = [L. .-lo] + [L. b]. and b is defined in (3.3) . We

have set 8 = CXZ;+ > 0. In the zero eigenvalue case. e = 0. Theorem A.2

yields for small A:

Setting again 19 = CA"''Olo yields (4.8).

We now estimate the remainder term R. Yotice that the same observation

as at the beginning of the proof of Proposition 3.11 shows that we have the

estimate

Therefore.

Now we have on V(.V):

where we recall that ( L - z ) - ' leaves V( X) invariant. Furthermore.

where i is defined in Proposition -4.2 (Section A.6). and I (u&g) is obtained

from I by replacing the form factor g by u&g. The last commutator in (4.10)

is hounded. and the other two are .V1/2-bonndecL so we obtain

Next. we estimate the first term on the r.h.s. of (4.9):

Combining this with (4.11) and (4.9). we arrive at the estimate

There is one more term in (4.S) we have to estimate: (iJ<iA),nca+w. Since P&V + ~ f ) P,I 2 0 and since pnfpn = 0. we haw the bound

Ii > p,lXipn + adj. 2 -CX.

which implies

Using (4.14) and (4.13). we obtain from (4.8)

xt. we we that for any 0. c > 0 (see Proposition .A.3 in Section .I.;).

Combining this with (4.15). *.ve obtain from (4.7):

On the last line. we used (4.1'1). We now choose the parameters as follo~vs:

then (4.16) is verified, and furthermore. (4.17) reduces to

On the other hand. recalling (4.6). we obtain by choosing the parameters v

and a as v = a3:

(recall that li7 = PkFA c. if e = 0). we obtain thus for small a from (4.18) and

(4.19) the relation

For e # 0. this is a contradiction. and it shows that there can not he any

eigenvalues of L in the intern1 A. Remark that there is no smallness con-

dition on the size of 1. except that it must not contain more than one

eigenvalue of Lo. so we can choose A = ( e - . e+ ).

Let us look now at the case e = 0. Again. we reach a contradiction from

(-1.20). provided

In this case. we conclude that zero is a simple eigenvalue of L. We show in

the next section that 11 Pn,,, P& 11 = O(3X). so if we take 3h small enough.

then (4.21) holds. and the proof of Theorem 2.1 is finished. H

Perturbation of the KMS state

In this section, we examine the perturbed IiMS state RjVA = U&A. where

U and were introduced in (2.22) and (2.21) respectively. We have. with

where RPJ is the particle Gibbs state. and R is the vacuum in 3( L2(R x S2)):

Here. Ll = uL~U-' is given by

Xotice that one can show essential selfacljointness of LI on DO exactly as in

the case for L. see Theorem 2.4. Remark also that for X = 0. LI reduces to

La and so RjmA reduces to RJ,o (recall that e-JL0/2RJ,o = R,s,o).

Theorem 4.1 (Perturbation of the K M S state). The vector RJVo is in

the domain of the unbounded operntor e-3L'12, for all 3 , and

uniformly in 3. A. provided JA 5 C fo r some constant C . In particular. w e

haue

The proof of Theorem 4.1 is based on the Dyson series expansion of

e-JL1/2flB,ot and it is given in [BFS4].

Appendix A

A . l Proof of Theorem 1.4

We use the notation introduced before Theorem 1.4 in Chapter 1. From

P( L~ = e , = C { i . , : E i J = ~ ) Pi :3 P,. we obtain together with the definition of

r ( e ) given in (1.9):

The idea here is to get a lower bound on the sum over (m. n ) E M x N by

summing only over a convenient subset of N x N (notice that every term in

the sum is positive). That subset is chosen such that the summands reduce

to simpler espressions.

Using the definition of rn (see ( 1.10)). we obtain

Summing over i. J and k. 1 according to (X. l) yields

For ( m . n ) E .\ir x A:. IW have P,.rn) = 0 and P,.(,, # 0. and for (n2.n) E 1 r

.\i;C x .\i;. we have P,.,,) # O m d P\l.lml = 0. .AS explained above. me now get ' t r

a lower bound on the sum ( A . 1 ) by summing only over the disjoint union

We obtain hence

where we used the fact that JE,,,n,c = 0 in these sums. In (A.2). we have

n E A:. so Em, = Em - En # e. Vm. Moreover, since rn E A:. then

Em - E, = e for some j . so we have Em, - c = E, - En # 0. Thus we obtain

and similarly

lu'ext. we investigate the integrals. From (2.25). (2.26). we get

uniformly in 13 2 1. With (1.18). (1.19) and remarking that o(CTC) = o ( T )

for any selfadjoint T. this yields

P ( L p = e ) r ( e ) P ( L p = e )

t E;, inf $0 ( E . , J s'2 d ~ ~ ~ ( ~ , , . o j ~ ' ) ( in f ~ E A G { d m ) + n inf E .trr {6,)

.A similar calculation as before yields

where we used in the last step (om. CG%on) = (0,. Go.). We hence get

Splitting the domain of integration W x St into R+ x S' u R- x S2 and using

(2.25) and (2.26). we arrive at

This together with (-4.4) gives

where we used J(E,,,, + c')p = S(E,,tn + s ) ( e - j E m n - I)- ' . Equation (-4.5) 1 / 2 shows that if we choose c, = Z i e-JEn/2. then each term in the sum is

zero. Recall now that the particle Gibbs state is given by

so (05, I I ( O ) R ~ ) = 0. Since T(0) 2 0. this implies that R; is a zero eigenvcc-

tor of r(0).

Finally me show that there is a gap in the spectrum of r(0) at zero. In-

deed. from (-4.5). me get by the definition of go (see statement of Theorem

2.2):

where we used En Icn l 2 = 1. Therefore. we obtain on Ran P;;: T ( 0 ) 2

2goZp(,d). This proves that if go > 0. then we have a gap at zero and zero is

a simple eigenvalue. I

A.2 Proof of Proposition 3.1

We first show that D(.\)nD(.V) is dense in 3. This implies that V( Lo)nV(.V)

is dense in R.

Let F(C,") denote the finite particle subspace of the Fock space 3(CF).

1.e.

F = { ( c - , ) , > ~ - E F(CF) s.t. cqn = 0 for dl but finitely many n}.

Clearly. F(C,X) c V(.\) n D(.V) c F, but F(CF) is dense in F ( L') which

is dense in 3. so F(CT ) is dense in 3.

Yext. for t E W. let Lt = let"^ . For cq E D(Lu) n D(X) = D(L,) n D( Ll) n V(.V). me have

t-l (e-t -40 L f e . O L ! ) , ( 1 1 , . . . . . U n ) = ( ~ ~ e ~ - ~ ~ a - ) ~ ( u l + t . . . . . (1, + t ) n

conclude that on V( Lo) n D ( S ) . e-'."~ L e t - 4 ~ = L I + t X. and since etaA0

commutes with L,. then

Next, let f E L 2 ( R x S2) . Then

(e-'""aa'( f)et-40q7) ( u l . . . . . u,) = (aa'( f ) f tA0 c), ( U I i f . . . . . U n + t )

= & P + f ( ~ . + t ) ( e ' - ~ ~ e * ) n-1 ( t r l + t ..... ~ r , - ~ + t )

= f i ~ + f ( t l ~ $ t ) h ' , - ~ ( ~ l ~ . . . . . U n - l )

= ( a m ( f t ) v ) , (u,. . . . . t i . ) .

where we defined the shifted function f '( u. a ) = f ( u + t . o) . and P+ is the

symmetrizatiou operator. We conclude that e-'-%'( f )&'o = a'( ft ) and

taking the adjoint also gives e-'.4~ a( f )et-"0 = cr( f t ). so

where & is obtained from I by replacing the form factors y by the translated

g t . gotice that f E L' + f t E L2 Vt. This shows the first part of Proposition

3.1.

Let us now take the derivative with respect to t . We have

Notice that if for some sequence { f,) in L2. f. + f in L2-sense. as s + 0.

then

This shows that +(& - I) -t i strongly on V(.V/?). providecl

which is equivalent to i&,I = i (as defined in Proposition 3.1). We finish

the proof of Proposition 3.1 by showing

Proposition A.1 With the conditions (1.8) (with p > 0 or p > 2). (A.8)

holds.

Proof. The calculations for g , and g2 are the same. We carry them out for

9,. From (2.25). we have

This gives

so for u f 0. and ,J > 1.

The IR and UV conditions ( 1.8) together with the regularity of g imply that

there is a 0 < R < x s.t.

Combining this with (A. 10). we obtain

We esamine the L2-convergence on the two regions of R x S2 defined by

(A.11) separately. First. we show that

as t + 0. and where we do not display the angle variable a. Since g, is in

C1 for IuI 3 R. we have from the Mean Value Theorem

where i ( u ) E (+I. Itl). Using (-4.11). we bound the square of the r.h.s. by

Clu + i (u ) l -*q 5 CIuI-'? uniformiy in t E (-1.1). Since /ul-?q is integrable

on {I ul 2 R). the Dominated Convergence Theorem shows (-4.12).

Xext, we show

(A. 13)

We cannot use the Mean Value Theorem directly. since Bugl has a singularity

at zero. However. this singularity is integrable. and me can deal with it by

considering a small neighbourhood of zero and the complement of it in the

domain of integration separately. as me explain now. Let

then for t r + 0. f t ( t r ) + l8,g1(u)l2 as t i 0. We want to show that ft i &gl

in L2i(-R. R ) x S?). Below. we give an integrable h l ( u ) s.t. gt( tr ) 5 h t ( t r ) .

and set . $ ht(u) -t J h(u) < x . A generalized version of the Dominated

Convergence Theorem then shows that fi + dugl in L2((- R. R ) x S') (see

for instance [F]. p.57. Exercise 20).

We now construct ht . Let v be a number s.t. ' < Y < 1. Since we P+ 1

are looking at the limit t i 0. we can assume that It 5 2 - k . so that

0 < It 1 5 It lu/2. On ( - R. If 1") x SZ. we have

9 l b + f - g l I 4 t = v u g , ) ( u + G(4)

In a similar way. we get for (u . a ) E (Itlu. R) x S2:

from (-4.11).

we have for ( t r . c r ) E (-R. R ) x s':

Xotice that for t # 0. both the 1.h.s. and the r.h.s. are integrable over

( -R . R ) x S'. Moreover. as t + 0.

Hence, if we can show that

as t -t 0. then the generalized version of the Dominated Convergence Theo-

rem sholvs (-4.13) as indicated above.

We thus finish the proof by showing (A.14). First.

as t + 0. This is obvious since we h o w the antideritative of u'p-'. In the

since -2 + Y + u ( 2 p + 1) > 0. so indeed (A.14) holds.


We have already shown in the proof of Proposition 3.1 that F ( C z ) is dense

in ( A ) 2 3 ) . Yow clearly F(C,Z) is dense in V(.-lo). so the density of

Do fl D( .v) n D( .ao) follows as in Proposition 3.1.

From Proposition 3.1. and the definition (3 .2) . we know that if L. E

Do D(J) n V(Ao). then

- & lim { ( P o - 1 U ' . L ~ ' . ~ ~ C * ) + ( LC*. Po - ,*) } . (A.15,

1 4 0 t t

By definition of .-lo as the generator of a strongly continuous group. we have etJ'0 - 1 for all c* E D(.&). l i ~ , - , 7 t* = .AoV. Moreover. we claim that k* -t

L c*. Indeed. since e-';'~ is unitary.

But since i ( L t - L) converges strongly on Do Do D(Y) (see Proposition 3.1).

then Lt - L + 0 strongly on that domain. Thus II(Lt - L)vl l + 0. as

t -+ 0. Sext, e - t " ~ - 1 + 0 strongly. since e-'s40 is strongly continuous. so

Il(e-t.40 - 1) Lull i 0 as t -t 0. Therefore we obtain indeed L e t - + ' ~ * -+ LC.

and so


Let us denote the spectrum of L, by a(Lp) = {ej}. where we include multi-

plicities. i.e. for degenerate eigenvalues. we have ej = er for different j # k.

Let Pj denote the rank one projector onto span{^^). where 9, E R, :j lip is

the unique eigenvector corresponding to el . Setting n2, = Pj m. we have

First. we estimate the term in the sum coming from {j : E, = e}:

;From our assumptions on g (see (1.8)) and (2.25). it is clear that l lg l /ul lLl =

C < m. uniformly in 13 1 1. LVe conclude that

Xest. we estimate

Xest. with the changes of variables y = u - ( e - e, ). we arrive at n

+I: dy (y2 + E ' ) - ' C [ i k j ( y + e - el. e) - f i j ( e - e,. e)]. e, #e

The mean d u e theorem yields for the last sum:

Using the Schwarz inequality for sums. we bound the modulus of the r.h.s.

from above by

We have to evaluate this at y = ij E (-(. <). Clearly. le - e, + ijI 2 le - e,I -

1j1 > do - E 2 do/2. if we choose F 5 d&. where

The r.h.s. of (A.24) can thus be estimated from above by

hence we arrive at

;From (-4.1 1 ). we know that the suprema are bouncied. uniformly in 3 2 1.

and so is 191 1. thus ( h.25) gives

. This argument Remark that the constant here depends on do. C - 8-'I2

is d i d for any p. Going back to the second term on the r.h.s. of (-4.21). we

have shown:

Xow me consider the first term on the r.h.s. of ( A ) . We see that. as

€/< -+ 0.

for any 0 < q < 1. This simply follows from the fact that for any such 0. we

have lim,,, r" Arctan(s) - 7-12) = 0.

x rj2Je - e,. L..) 4 u - ( e - e,))m,(u. c*)

e, f e

We conclucle that

Choose e.g. f = el/.' and q close to 1. then the above r.h.s. is

This together with (-4.18) yields

A.5 Proof of Remark 3 after Theorem 1.1

We show that for g E CF. g(L) leayes 'D(.V'I~) invariant. Using the repre-

sentation for 0 < p < 1 (see for esample [I<]. Chapter V. end of Paragraph 3

(p.2S6)) :

sin ap " (\. + l ) - P = - 1 x-~(-V + 1 + ~ ) - l & .

X

we obtain

Here. we used 1. of Proposition. Clearly. it is enough to show that the range

of the second term lies in D(.Y'/'). Let us rewrite W2 applied to the second

term as

where we defined for q > 0:

It is easy to see that IITvli 5 C. uniformly in q > 0: indeed. the norm of

the integrand can be bounded above for small r by x-'I2. and for large x

by s-'-9. hence it is integrable. The purpose of introducing rl is that for

q < 112. we have now only to consider in (-4.29)

where p = 1 - n is now strictly bigger than 112. Csing again (A.2S). we write

and we split the domain of integration into two pieces. s 2 IIm-I-' and its

complement. On the first piece. me obtain

where D(:. x ) is a bounded operator satisfying IID(z. s)ll 5 C. This follows

from 1. of Proposition. and since x 2 IIm: I-*. For large r . the norm of the

integrand behaves as s-p-'f2. which is integrable since p > 112. The whole

integral is therefore hounded above by C I I n c /'P-'.

Xext. we treat the other piece in the integral over x:

where the positive numbers s, are chosen to satisfy s, 2 IImf2. so that

11 D( r. sr ) 1 1 5 C (as above. this follows from 1. of Proposition). The norm of

the integrand is bounded above by ~ s - p s ~ ~ ' ~ ( 2 + s, ) lIrnzld1. We conclude

that 11(.4.31)1( is bounded above by C( l + IIxn=I2~-'l). Combining this with

the estimate for the integral over x for large I. we conclude that

which gives

This finishes the proof.

A.6 Properties of some commutators

The results of the following proposition are i.1scc1 in the proof of Theorem 1.2.

see Section 4.1.

Proposition A.2

1. Let z E C\W, s > 0. Then we have

2. Let g E C,Z. s > 0 and I as in 1. above. Then we have on D ( L ) :

9. Let h E C 5 . t . Ih(')(x)I 5 CI-\ for large .r . P = 0 .1 .2 .3 . On

V(L) n V ( X ) = V(L,) n V(.V), w e have for r > 0:

1 Lh( i&/ r ) = h ( i & / r ) L - h t ( i & / r ) - X + T ( r . A ) . (A.33)

r

,where the t e r n T ( r . A) = A[I. h(i.-lo/r)] satisfies IIT(r. x)(.v+ l)-'/'II 5

C A / r and ll(.\' + 1)- ' l4T(r . A ) ( N + 1)-11411 5 CX/r, and furthermore.

Proof. We feel that careful proofs of these and similar (formally more or

less obvious) formulas are rarely seen in the litterature. and hence we give

detailed proofs.

Let us start with the proof of 1. For notational convenience. we write

simply L- ' . L i l . 3-' for ( L - z ) - ' . ( L o - -)-'. (.V+s)-I. if z or s is irrelevant

for the computation. We have

where w e were allowed to separate L-I from L .Ye' L-' . since

as rve show now. Indeed. on D(.\'). we have

where L-' and L L,' can be separated since L,'V(.V) = D( Lo) fl V(.V) =

D(L) n D ( S ) . S o n on V(Lo) n 'D(.V''). L splits as Lo + .\I. so we obtain

on V(.V):

This implies that on the whole space. we have

where the second term on the r.h.s. is bounded. Taking the adjoint of the

last expression gives

which shows that Ran iV-' L- ' c Z)( Lo ). Xow since we have clearly also

Ran N- ' L - ' c Z)( iV ) . then

which shows (-4.35).

Next. we have La\;-' L-I = La.V-'L-' + AI.V-lL-L. and

which gives. together with (-4.34) and restoring the z and s :

Nest. it is not difficult to explicitely calculate

Using this relation and its adjoint to commute (.\i + s ) - ' I 2 through I in

(A.36). we obtain the desired result.

Let us now prove 2. Take f E CF(4 . l ) s.t. f ( 0 ) = 1. f t (0 ) = 0. Let

r > 0. then me haw strongly on D(L): lim,,, r F ( L / r ) + L. We calculate

the commutator

lim r [F(L/ r ) . g(.Y/s)]iq r -+s

(which is easily seen from the where ~ v e used s-lirn,,,x(L/r - z ) - ' = - - - I

spectral theorem). Xotice that the integrals over 2 and (' have split. and

J d ~ ( z ) z - ? = F f ( 0 ) = 1. .kwune now cq E D(L). and write out the comrnu-

tator to see that

lim rF(L/r)g(.V/s)c* = lirn f(L/r)Ly(.V/sls)c* r+30 r + s

exists and is equal to

The result now follows since if for some r i t . f ( L / r ) La* + eq. then c. E V(L ).

and L y = (this follows from the spectral theorem).

?low we prove 3. We start by showing the following relation on V( Lo) n V( .V) :

where sgn(x) = f 1 according to whether x > 0 or r < 0. To prove (-4.38).

notice that the spectrum of .-lo is the imaginary axis. and so for Re: > 0. we

have

hence c 4 o S maps V( L o ) n V(.V) into V( L o ) . and since .-Lo commutes with S.

C'O" leaves D( Lo) n D(.Y) = P( L) n D ( S ) invariant. We obtain therefore

from (-4.39) on this invariant domain:

This. together with a similar calculation for z with Re: < 0 yields (-4.38).

Xext. we prove (h.33) first for h E Ci. then we mill show it holds also for

h as indicated in the proposition.

Using (-4.3s): we calculate on D(L) n V(N) :

i = h(i.-l0/r) L - - h ' ( i & / r ) S + T ( r . A ) .

r

where we used that sgn(Re r z l i ) = sgn(r1mr) = sgn(Irn2). Here. T ( r . X ) is

given by the iterated integral on the r.h.s:

Xotice that

uniformly in s > 0. Indeed. for large s this is clear since 1J.V + I)-'/? and

I ( X + 1)-'I2 are uniformly bounded in s. and for small s. it is true since

I , ( X + 1)-'I2 is differentiable at s = 0. From this. we conclude that

The second statement in 3. about T is proved just like the last estimate. We

now prove the statement about convergence.

With a change of variables a = rs in the s-integral. me obtain from (A.40):

Integrating by parts in the a-integral gives

The limit of the last integral as r -t x is - z - ? . This shows that

lim r T ( r . X ) ( X + I)-'/' = -iXht(0)l(&g)(.V + I)-'/'. r -KG

where me used J d L ( ~ ) z - ~ = h t (0 ) .

We finish the proof by showing (h.33) for functions h E C3 with the

bounds indicated in the proposition. Let 1 E Ci be a smooth characterisitc

function of an interval around zero. and set for rn > 0: h , ( x ) = h ( x ) i ( r / r n ) .

Clearly. h , is in C;. so (A.33) holds for h , :

t Lh, = h , L - hk -?I + T,,.,(r. A ) .

r

where Tm is obtained from T by replacing h by h,. It is clear that in

the strong sense. as rn + r;: hm + h and h;ix) = h ~ ( . r ) ~ ( x / m ) + h(x)\ ' (x/m)/rn + h l ( x ) . and one also shows that T,(r. A ) + T ( r , A ) (for

a more general procedure of this kind. see e.g. (HS31). We conclude that

tjb E D(L) n D(.v):

where the r.h.s. is well defined. Furthermore. since h , c* + h c?. and L is

closed. we have

Proposition A.3. The following equality holds in the sense of operators

on D(S1l2) or in the sense of q.uad~ut2c f o m ? on V(.V'/.'):

1 [ I . i fa] = fLadf,o ( I ) - f:adi, ( I ) + R.

"

where we a s v m e that the k-fold cornm~rtator

is i~' l ' -bounded ( o r ~ ' l ~ - f o r m bounded) for k = 1.2.3. The t e r m R satisfies

the estimate

Proof. LXng the functional calculus introduced in Appendix B.2. we

write

[I. i f = 1 dj(z)(ic1-4~ - : ) - ' [ I . .-lQ](ia& - :)-'

L = fLnd\, ( I ) - 7 3 f:n@,, ( I ) -

-a2 J cij(:)(ia~~ - 2 ) % d i 0 ( ~)(ia~~ - z ) - ~ .

The last integral is defined to be R. and the estimates follow by noticing that

.Ao and commute. I

A.7 O n a smooth partition of unity

We show a result about a smooth partition of unity used in the proof of

Theorem 1.2. Section 4.1.

Proposition A.3

1) There is n CS-partition of unity

s.t. fi(x) = 1 for x 5 -1. f l ( r ) = 0 for s 3 1. f l 2 0. f[ 5 0 and s.t.

V r > 0. V p > 0:

#(.)I 5 E f i ( ~ ) + ~-'-~f2(x). x E R.

2) There is a C"-partition of unity

g:(z) + g,Z(x) = 1. x E R.

Remaark. By afine transformation of the coordinate s rt ax + 6. one can

realize two families of part it ions of unity satisfying ( .1.41) and (-4 .E) respec-

tively and having variable steepness of the descending/ascending flanks at a

variable position.

Proof. 1 ) W e coustruct f 1,2 esplicitcly. Set (modulo a normalization con-

stant)

J \ - 1

O else.

and clcfine

Then. f2 E Cr . and f i = 1 - fi has all the properties indicated in the

proposition. The only non-trivial property to check is (-4.41 ). Since Iji (x) I =

I f&r) 1 = &(x) = f (s). it is enough to show

for C, < CE-'-P. Let I, = {r E R : f (x ) 2 r l } . Then clearly. for x E I:

(complement ). (A.13) is satisfied (for small c). We want now to find C: =

C, - E s.t. (X.43) is also satisfied for x E I,. so

f ( 4 c: = sup - f 2 ( ~ )

does the job. Notice that fi > 0 on I,. Taylor expanding f2 around r , l z (the

left endpoint of the interval ICl2) and using the Mean Value Theorem. it is

not difficult to estimate the supremum from above by C E - ' - ~ . for all p > 0.

This shows 1).

To prove 2) . we notice that by setting

me get again a Cr-partition of unity s.t. now

One calculates

which gives. using f,? + fj 2 112 and 1):

19; 1 5 23/21f;l 5 25/2Ef1 + 2 " 2 ~ , f 2 .

Since f i .2 = Jmgl,? 5 91.2. this yields

lgi 1 1 23'2~gl f 2 3 1 2 ~ , g 2 .a

Appendix B

B . l Fock space

For a more detailed esposition of the material of this section. we refer to

[BRII] or [GI].

Fock space, vacuum, second quantization. Let 4 be a one-particle

Hilbert space. e.g. 4 = L2(R3. d3h) . for scalar Bosons without spin. The

Hilbert space describing a sys tern of .V (identical and indistinguishable) Bose

particles is the Xfold symmetrized tensor product P+fj . . 2 f j (in the

example above. this is the space of all symmetric square integrable functions

of .V variables kl . . . . . ks. each in W3). We introduced the symmetrization

operator P+. which is a projection. defined on a dense set as

V fl . . . . . fiv E 8. and where the sum runs over all permutations of (1. . . . . J). In order to incorporate the possibility of creation and annihilation of par-

ticles. we introduce the (Bosonic) Fock space

where we understand P+Qa0 = @. An element of Fock space can be repre-

sented by a sequence t,!~ = ($vn)r=5=(1, where t L v n E P+fian is said to be in the

n-sector. s.t. I l + l l r = xr.=o [ld~nll~+oa,. , < t m . Yotice that the construction of

Fock space is based on the notion of -number of particles S*'. which becomes

an observable. i.e. an (unbounded) selfadjoint operator on F. defined by

where the subscript n means we take the projection onto the rr-sector. i.e.

V o E 3, on is the n-th element in the sequence {on):==, representing n. By

definition of 3 and .V. .V is selfadjoint on the dense domain

There is a distinguished vector in Fock space. called the vacuum R. given

by Ro = 1. R, = 0. V n > 0. LYe see that .VR = 0. and this is why R is

interpreted as the no-partide state. the vacuum.

The particular structure of Fock space allows us to lift the action of

any operator .4 on f) to an action on 3. This procedure is called second

quantization, it is the map .-I H dl?(.-l). where dr(.-L) is defined on a dense

set as

V fi.. . . . f n € D ( 4 . and then estended by linearity to ( a subspace of) Fock

space. We also define d r ( 4 = 0 on @. Moreover. for -4 acting on 4. define

the operator r(A) acting on the n-sector by

By linearity. r(.4) acts on Fock space. Xotice the formal relation T(8) =

edr(--u

Creation, annihilation operators, Weyl operators, CCR. Yext.

we introduce the creation and annihilation operators:

V f E 4. which yields by linear extension two well-defined operators on

D ( N ' / ~ ) . with

where a# denotes either creator or annihilator. Creators and annihilators

are adjoints to each other in the sense that Vc. o E D(.VL/'). V f E 4:

Yotice that f ci a'( f ) is linear. while f ct a ( f ) is antilinear. Also. it is

clear that am( f ) sends the n-sector into the ( n + 1)-sector. hence it creates

a particle. and similarly, a( f) destroys a particle. Without difficulty. one

shows that the following CCR (Canonical Commutation Relations) hold:

where the scalar product refers to the Hilbert space Sj.

We can decompose a( f) into its real and imaginary parts by introducing

the field operators

so that a( f) = 2-'/'(+( f ) + in( f ) ) and a'( f) = 2-L'2(c~( f ) - in( f ) ) . One

shows that $( f ) is essentially selfadjoint on the finite particle subspace. that

the linear span of

is dense in F. and that on V(.V). one has

For a proof of these assertions. see e.g. [BRII]. Proposition 5.2.3. We identify

@( f ) with its selfadjoint closure. and define the Weyf operators

which is a unitary operator on .?. The advantage of the Weyl operators is

that they are bounded. in contrast to their generators. the field operators.

From the CCR (B.2). it follows that Vf.g E 4:

which is called the Weyl-form of the CCR. Sotice that {Me*( f ) : f E a} is a *-algebra. but it is not closed in the operator norm topology on B ( 3 ) .

hence it fails to be a C'-algebra. We define the Weyl CCR-algebra M?(fi)

to be the Cm-algebra generated by {W( f ) : f € 8). In other words. W ( 4 )

is the smallest C'-algebra containing {W( f ) : f E 4). We define the Fock

representation of the Wejl CCR-algebra to be the pair (3. rF ) . where

is a *-homomorphism. and 3 is the Fock space over S j . One shows that

since the span of ( B . I ) is dense. then { I T ( f ) : f E 4) is irreducible on 7

(see [BRII]. Proposition 5.2.4). so every nonzero vector in 3 is cyclic ( [BRI].

Proposition 2.3.8). This means that any normalized c E 3 defines a state

( a normalized. positive. linear fnnctional on W ) . called a vector state. by

s,. : w -t c. ;,.(.A) = ( L ' . - 4 4 .

hence gives rise to a cyclic representation (3. TF. cq) of (W.+ ) . Each such

cyclic representation is a concrete realization of ( i.e. is unitarily equivalent

to) the GXS representation of (W.d , . ) . The Fock state s p is defined as

where again. R is the vacuum in 3. Xotice that

is dense in 3 ( this follows directly from the definition of the creation opera-

tors)? and that for any normalized (1" = C a w ( fl). * . a'( f n )R . where C E @ is

a normalization factor. we have

so for such e. + can always be expressed in terms of the Fock state.

The great advantage of the Fock state is that Wick's theorem holds, i.e.

the average of the product of any (finite) number of creators/annihilators.

(0. ny=, a#( f ; ) ~ ) . can be expressed as the sum over products of terms

(n. a ( f i )a'( f, )n). In other words. the pair correlation functions completely

determine the Fock state.

B .2 Operator calculus

We out line an operator calculus for functions of selfadjoint operators. used

extensively in this work. For a detailed exposition and more references. we

refer to [HS3].

Let f E Ct(R). I: 2 2. and define the compactly supported complex

measure

where z = r + iy and f is an almost analytic complex extension of f in the

sense that

Then. for a selfadjoint operator -4. one shows that

where the integral is absolutely convergent. Given f . one can construct es-

plicitely an almost analytic extension f supported in a complex neighbour-

hood of the support of f . One shows that for p 5 k - 2.

where

and (x) = ( 1 + x 2 ) ' / ' . Furthermore. the derivatives of f ( A ) are given by

We finish this outline by mentioning that these results extend by a limiting

argument to functions f that do not have compact support. as long as the

norms in the r.h.s. of (B.3) are finite.

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