# Correlation decay and Markovianity
in open systems

Marco Merkli
Department of Mathematics and Statistics
Memorial University of Newfoundland
St. John’s, Canada, A1C 5S7

###### Abstract

A finite quantum system ${\rm S}$ is coupled to a thermal, bosonic reservoir
${\rm R}$. Initial ${\rm S}{\rm R}$ states are possibly correlated, obtained
by applying a quantum operation taken from a large class, to the uncoupled
equilibrium state. We show that the full system-reservoir dynamics is given by
a markovian term plus a correlation term, plus a remainder small in the
coupling constant $\lambda$ uniformly for all times $t\geq 0$. The correlation
term decays polynomially in time, at a speed independent of $\lambda$. After
this, the markovian term becomes dominant, where the system evolves according
to the completely positive, trace-preserving semigroup generated by the Davies
generator, while the reservoir stays stationary in equilibrium. This shows
that (a) after initial ${\rm S}{\rm R}$ correlations decay, the ${\rm S}{\rm
R}$ dynamics enters a regime where both the Born and Markov approximations are
valid, and (b) the reduced system dynamics is markovian for all times, even
for correlated ${\rm S}{\rm R}$ initial states.

##  1 Introduction

Open quantum systems are most commonly described by a tensor product Hilbert
space ${\mathcal{H}}_{\rm S}\otimes{\mathcal{H}}_{\rm R}$ (system and
reservoir) and an interacting Hamiltonian

$H_{\lambda}=H_{\rm S}\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm
R}+\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}_{\rm S}\otimes H_{\rm R}+\lambda V$

where $H_{\rm S}$ and $H_{\rm R}$ are the system and reservoir Hamiltonians,
$\lambda$ is a coupling constant, which we consider to be small in a suitable
sense, and $V$ is an interaction operator. In the literature, coupled ${\rm
S}{\rm R}$ systems are sometimes studied when the parts ${\rm S}$ and ${\rm
R}$ are of comparable size. In the current work, we focus on the situation
$\dim{\mathcal{H}}_{\rm S}=N<\infty$ and $\dim\mathcal{H}_{\rm R}=\infty$, and
a reservoir Hamiltonian $H_{\rm R}$ with continuous spectrum. In this sense,
the reservoir ${\rm R}$ is much larger than the system ${\rm S}$ in our
considerations. The dynamics of an initial state (density matrix) $\rho_{{\rm
S}{\rm R}}$ is given by the Liouville–von Neumann (or Schrödinger) equation

$\rho_{{\rm S}{\rm R}}^{t}=e^{-{\rm i}tH_{\lambda}}\rho_{{\rm S}{\rm
R}}\,e^{{\rm i}tH_{\lambda}}$

and the reduced system density matrix is obtained by taking the partial trace
over the reservoir,

$\rho_{{\rm S}}^{t}={\rm tr}_{\rm R}\,\big{\\{}e^{-{\rm
i}tH_{\lambda}}\rho_{{\rm S}{\rm R}}\,e^{{\rm i}tH_{\lambda}}\big{\\}}.$

Uncorrelated initial states.  In textbooks and research articles, most often
the assumption is made that the initial state is uncorrelated, that is, of the
product form $\rho_{{\rm S}{\rm R}}=\rho_{\rm S}\otimes\omega_{\rm R}$. Then
one can define the dynamical map acting on system density matrices,

${\mathcal{V}}_{t}:\rho_{\rm S}\mapsto\rho_{\rm S}^{t}.$

$\mathcal{V}_{t}$ is the flow or propagator of the system dynamics and of
course, it depends on the initial reservoir state $\omega_{\rm R}$. For
example, if $\omega_{\rm R}=\omega_{{\rm R},\beta}$ is the reservoir
equilibrium state at a given temperature $T=1/\beta$, then of course the
system dynamics, and hence $\mathcal{V}_{t}$, will depend on $T$. Due to the
interaction of ${\rm S}$ and ${\rm R}$, mediated by the operator $V$, the
state $\rho_{{\rm S}{\rm R}}^{t}$ is generically correlated (not of product
form) for $t>0$. This results in the violation of the group property,

$\mathcal{V}_{t+t^{\prime}}\neq\mathcal{V}_{t}\mathcal{V}_{t^{\prime}}.$

If the reservoir is not much influenced by the coupling to the system and
initially in a stationary state under its own dynamics, then one might
intuitively expect that $\rho_{{\rm S}{\rm R}}^{t}\approx\rho_{\rm
S}^{t}\otimes\omega_{\rm R}$. This is called the Born approximation. If on
top, one assumes that the reservoir loses its memory quickly, then one may
expect the Markovian property to hold approximately,

$\mathcal{V}_{t+t^{\prime}}\approx\mathcal{V}_{t}\mathcal{V}_{t^{\prime}}\quad\mbox{or}\quad\mathcal{V}_{t}\approx
e^{t{\mathcal{L}}},$

where the generator ${\mathcal{L}}=\mathcal{L}(\lambda)$ depends implicitly on
the initial reservoir state $\omega_{\rm R}$ and $\mathcal{L}(0)=-{\rm
i}[H_{\rm S},\cdot\,]$ generates the uncoupled system dynamics. It has been
known since the seventies (see the founding papers [10, 11, 41]) that in some
generality, the Markovian approximation is valid in the weak coupling, or van
Hove limit. This means that

$\lim_{\lambda\rightarrow
0}\sup_{0\leq\lambda^{2}t<a}\|\mathcal{V}_{t}-e^{t{\mathcal{L}}_{\rm
S}(\lambda)}\|=0\qquad\mbox{for any $a>0$},$ (1.1)

where the superoperator

${\mathcal{L}}_{\rm S}(\lambda)={\mathcal{L}}_{\rm
S}+\lambda^{2}{\mathcal{K}},\qquad{\mathcal{L}}_{\rm S}=-{\rm i}[H_{\rm
S},\cdot\,]$ (1.2)

is called the Davies generator. It generates the CPTP semigroup
$e^{t\mathcal{L}_{\rm S}(\lambda)}$ [9, 16, 29] (completely positive trace
preserving). The operator $\mathcal{K}$, a sum of a Hamiltonian plus a
dissipative term, describes the influence of the bath to second order in
perturbation [1, 7, 36]. From (1.1) we have an alternative expression for
$\mathcal{K}$,

$\lim_{\lambda\rightarrow 0}\
\mathcal{V}_{\tau/\lambda^{2}}\,\circ\,e^{-\tau{\mathcal{L}}_{\rm
S}/\lambda^{2}}=e^{\tau\mathcal{K}},\qquad\forall\tau\geq 0.$ (1.3)

The shortcoming of (1.1) is that it guarantees the accuracy of the Markovian
approximation only for bounded values of $\lambda^{2}t$. Said differently,
(1.1) shows that the Markovian approximation is guaranteed to hold for
$t\rightarrow\infty$ only if at the same time, one takes $\lambda\rightarrow
0$ in a way such that $\lambda^{2}t$ stays finite. This deficiency of (1.1)
was recently overcome in [24, 29, 30, 31], where it is proven, under suitable
assumptions, that

$\sup_{t\geq 0}\big{\|}\mathcal{V}_{t}-e^{t({\mathcal{L}}_{\rm
S}+\lambda^{2}\mathcal{K})}\big{\|}\leq C|\lambda|^{1/4},$ (1.4)

provided $|\lambda|\leq\lambda_{0}$ for some $\lambda_{0}>0$. Due to (1.4),
the Markovian approximation is now guaranteed to be accurate to
$O(|\lambda|^{1/4})$ uniformly in time $t\geq 0$, for all small, fixed
$\lambda$. The error bound $|\lambda|^{1/4}$ is derived for the least amount
of regularity of the interaction operator necessary for the proof – under a
more stringent (analyticity) assumption, it can be improved to $|\lambda|^{2}$
(see [24]). In the recent work [8] it is shown that the validity of the
Markovian approximation for arbitrarily large finite times does not imply the
validity for all times. There the authors consider the zero temperature,
dissipative Janes-Cummings model in the singular coupling limit. Given any
time $t_{0}>0$, they explicitly design SR interactions (form factors $g(k)$)
in such a way that the system (qubit) dynamics is exactly Markovian up to
$t_{0}$, but for $t>t_{0}$ the dynamics becomes non-Markovian. Even though
this example uses the singular coupling limit ($g(k)$ not square integrable),
the result indicates that the passage from (1.1) to (1.4) is not a mere
technicality.

Correlated initial states. The assumption of initially uncorrelated SR states
may not be physically realistic. It requires to bring together, at time $t=0$,
a system and a reservoir which at earlier times did not interact (or which by
some measurement or dynamical fluke would happen to be in a product state).
One should then ask about the dynamics of correlated initial states, that is,
ones which are not of product form, see [17, 33, 37, 38, 39] and the
references in there. For a given correlated initial state $\rho_{{\rm S}{\rm
R}}$, the reduced system dynamics $\rho_{\rm S}^{t}$ is still well defined as
above. However, one cannot define the dynamical map $V_{t}$ any longer, as
different $\rho_{{\rm S}{\rm R}}$ can have the same marginal $\rho_{\rm S}$
(say, due to different ${\rm S}{\rm R}$ correlations). Those different initial
${\rm S}{\rm R}$ states evolve to states at time $t>0$ which have different
system marginals, which means that the initial marginal cannot be mapped
consistently to the marginal at a later time. In our investigation, we start
with a correlated initial state $\rho_{{\rm S}{\rm R}}$ and describe its well
defined reduced system dynamics $t\mapsto\rho^{t}_{\rm S}$. The opposite
approach is also studied in the literature, starting with an initial system
state $\rho_{\rm S}$, then mapping it to a ${\rm S}{\rm R}$ state by an
assignment map $\rho_{\rm S}\mapsto\rho_{{\rm S}{\rm R}}$. The initial
correlation is then encoded in the assignment map, whose structure can be
analyzed [2, 14, 15, 35, 40].

Outline of main results.  We consider a class of open systems for which the
Markov approximation (1.4) for uncorrelated initial states, $\rho_{\rm
S}\otimes\omega_{{\rm R},\beta}$, was proven in [31]. Here, $\omega_{{\rm
R},\beta}$ is the thermal equilibrium state of the reservoir at temperature
$T=1/\beta>0$.111This is not a necessary feature for our approach to work, one
may also take non-equilibrium stationary states, as we will explain elsewhere.
Our main result, Theorem 2.2, gives an expansion of the full ${\rm S}{\rm R}$
dynamics with correlated initial states $\rho_{{\rm S}{\rm R}}$. It can be
expressed as follows – see after Theorem 2.2 for more in-depth and precise
statements. We show that

$\rho^{t}_{{\rm S}{\rm R}}=\big{(}e^{t{\mathcal{L}}_{\rm S}(\lambda)}{\rm
tr}_{\rm R}\\{\rho_{{\rm S}{\rm R}}\\}\big{)}\otimes\omega_{{\rm
R},\beta}+\chi(t,\lambda)+O(|\lambda|^{1/4}),$ (1.5)

where $\mathcal{L}_{\rm S}(\lambda)$ is the Davies generator (1.2) and
$\chi(t,\lambda)$ describes the dynamics of the ${\rm S}{\rm R}$ correlations.
The relation(1.5) is valid for all $|\lambda|\leq\lambda_{0}$, some
$\lambda_{0}>0$, with a remainder independent of time $t\geq 0$. It is to be
interpreted in a weak sense, that is, when applied to operators from a ${\rm
S}{\rm R}$ observable algebra $\mathcal{O}$, (2.19). The initial states are
taken from a class of states obtained by applying (very general) quantum
operations on the uncoupled ${\rm S}{\rm R}$ equilibrium state $\rho_{{\rm
S},\beta}\otimes\omega_{{\rm R},\beta}$ (see (2.23)). We show the following
results:

  * –

For $\rho_{{\rm S}{\rm R}}=\rho_{\rm S}\otimes\omega_{{\rm R},\beta}$ (with
$\rho_{\rm S}$ any system density matrix), we have $\chi(t,\lambda)=0$. This
shows that the Born approximation holds and recovers the Markovianity result
(1.4).

  * –

$\chi(t,\lambda)$ also vanishes when applied to observables $O_{\rm
S}\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm R}$ of ${\rm S}$ alone, for any
initial $\rho_{{\rm S}{\rm R}}$. This means that the initial correlations only
influence the dynamics of observables which involve the reservoir. In
particular, this shows that the Markovian approximation is also valid for
correlated initial states, and for all times $t\geq 0$.

  * –

In the presence of initial correlations, the Born approximation ($\rho_{{\rm
S}{\rm R}}^{t}\approx\rho_{\rm S}^{t}\otimes\omega_{{\rm R},\beta}$) is
obviously not correct for small times. However, we show that
$\chi(t,\lambda)\sim 1/(1+t^{3})$, which implies that initial correlations
decay, and after that, the ${\rm S}{\rm R}$ state enters a regime where the
Born approximation is valid, even for correlated initial states. This is a
statement about the full ${\rm S}{\rm R}$ state dynamics. In contrast, as
explained above, the reduced system state is approximated by the Markovian
semigroup for all, even small, times.

##  2 Model and main results

###  2.1 Model

We consider an $N$-level system interacting with a reservoir modeled by a
continuous family (field) of quantum harmonic oscillators, and we will largely
follow the notation of [30]. The system and reservoir Hamiltonians are given
by

$H_{\rm S}=\sum_{j=1}^{N}E_{j}|\phi_{j}\rangle\langle\phi_{j}|\mbox{\quad
and\quad}H_{\rm R}=\int_{{\mathbb{R}}^{3}}\omega(k)a^{*}(k)a(k)d^{3}k$ (2.1)

respectively, acting on the Hilbert space

${\mathcal{H}}_{{\rm S}{\rm R}}={\mathbb{C}}^{N}\otimes{\mathcal{F}}.$ (2.2)

We often simply write $H_{\rm S}$ for $H_{\rm S}\otimes\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}$
and so on. The $E_{j}$ are the system energies and $\omega(k)\geq 0$ is the
frequency of the mode $k$ ($\hbar=1$), which we take for definiteness to
follow the dispersion e.g. of photons,

$\omega(k)=|k|.$ (2.3)

The energy eigenstates $\phi_{j}$ form an orthonormal basis of the system
Hilbert space ${\mathbb{C}}^{N}$ and the (momentum representation) creation
and annihilation operators satisfy the canonical commutation relations
$[a(k),a(\ell)]=0=[a^{*}(k),a^{*}(\ell)]$ and
$[a(k),a^{*}(\ell)]=\delta(k-\ell)$ (Dirac delta $k,\ell\in{\mathbb{R}}^{3}$).
The reservoir Hilbert space is the (Bosonic) Fock space

${\mathcal{F}}=\bigoplus_{n\geq 0}\,L_{\rm
sym}^{2}({\mathbb{R}}^{3n},d^{3n}k),$ (2.4)

built over the single-particle space $L^{2}({\mathbb{R}}^{3},d^{3}k)$, the
square integrable scalar valued functions. The full, interacting Hamiltonian
is given by

$H_{\lambda}=H_{\rm S}+H_{\rm R}+\lambda G\otimes\varphi(g),$ (2.5)

where $\lambda\in\mathbb{R}$ is the coupling constant, $G^{*}=G$ is any
hermitian system matrix and

$\varphi(g)=\frac{1}{\sqrt{2}}\big{[}a^{*}(g)+a(g)\big{]},\quad
a^{*}(g)=\int_{{\mathbb{R}}^{3}}g(k)a^{*}(k)d^{3}k,\quad a(g)=[a^{*}(g)]^{*}$
(2.6)

are the field, creation and annihilation operators, respectively. The function
$g(k)\in L^{2}({\mathbb{R}}^{3},d^{3}k)$ in the interaction (2.5) is called
the form factor – generally, when we consider $a^{*}(f)$, $a(f)$ we also call
$f$ test functions.

The above is a paradigmatic model for open quantum systems. For $N=2$ it is
called the spin-Boson model. As presented here, it is the continuous mode (or
thermodynamic, infinite-volume) limit of a system described by a discrete set
of oscillators with Hamiltonian

$H_{\rm R}^{\prime}=\sum_{k}\omega_{k}a^{*}_{k}a_{k}$ (2.7)

and field operator in the interaction given by
$\varphi^{\prime}(g)=\sum_{k}g_{k}a^{*}_{k}+{\rm h.c.}$ In the theoretical
physics literature, the limit of continuous values of $k$ is often performed
at the end of calculations of various expressions, such as transition
probabilities. In this limit of a continuum of reservoir modes one can analyze
irreversible dynamical effects (like time decay). In the present work, we
start off directly with the reservoir having a continuum of frequencies,
(2.1).

In contrast to the common assumption of initially uncorrelated system-
reservoir states of the factorized form $\rho_{\rm S}\otimes\rho_{\rm R}$, we
are concerned here with initial states obtained by correlating the system with
the reservoir. The system Gibbs equilibrium density matrix at temperature
$T=1/\beta$,

$\rho_{{\rm S},\beta}=Z_{{\rm S},\beta}^{-1}\,e^{-\beta H_{\rm S}},$ (2.8)

$Z_{{\rm S},\beta}={\rm tr}_{\rm S}e^{-\beta H_{\rm S}}$, is a well defined
density matrix. One can represent $\rho_{{\rm S},\beta}$ as a vector state
$|\Omega_{\rm S}\rangle\langle\Omega_{\rm S}|$ in the enlarged Hilbert space
${\mathbb{C}}^{N}\otimes{\mathbb{C}}^{N}$ as follows. For any $O_{\rm
S}\in{\mathcal{B}}({\mathbb{C}}^{N})$ (bounded operators),

${\rm tr}_{{\mathbb{C}}^{N}}\big{(}\rho_{{\rm
S},\beta}A\big{)}=\langle\Omega_{\rm S},\pi_{\rm S}(O_{\rm S})\Omega_{\rm
S}\rangle_{{\mathbb{C}}^{N}\otimes{\mathbb{C}}^{N}},$ (2.9)

where $\pi(O_{\rm S})=O_{\rm S}\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{{\mathbb{C}}^{N}}$
and $\Omega_{\rm S}\in{\mathbb{C}}^{N}\otimes{\mathbb{C}}^{N}$ is a normalized
vector (explicitly given in (3.7)). This purification representation is
sometimes very helpful. For the continuous mode reservoir the situation is a
bit more cumbersome. Its equilibrium state cannot be expressed as a density
matrix acting on $\mathcal{F}$, because $e^{-\beta H_{\rm R}}$ is not trace
class ($H_{\rm R}$ has continuous spectrum). Rather, the reservoir equilibrium
state is constructed as the limit of equilibrium states $\propto e^{-\beta
H^{\prime}_{\rm R}}$, where $H^{\prime}_{\rm R}$, (2.7) is trace class. The
limit state is expressed as a positive, linear functional $\omega_{{\rm
R},\beta}$ on reservoir operators as follows. Let

$L^{2}_{0}=L^{2}({\mathbb{R}}^{3},d^{3}k)\cap
L^{2}({\mathbb{R}}^{3},|k|^{-1}d^{3}k).$ (2.10)

For $f,g\in L^{2}_{0}$ we have [6, 32]

$\omega_{{\rm R},\beta}(a(f))=\omega_{\rm{\rm
R},\beta}(a^{*}(f))=0,\quad\omega_{{\rm
R},\beta}\big{(}a^{*}(f)a(g)\big{)}=\langle g,(\,{\rm
e}^{\beta|k|}-1)^{-1}f\rangle,$ (2.11)

where $\langle f,g\rangle=\int_{{\mathbb{R}}^{3}}\bar{f}(k)g(k)d^{3}k$.
Expectations of arbitrary polynomials in creation and annihilation operators
are calculated using Wick’s theorem [6] and the expecation of a unitary Weyl
operator

$W(f)=e^{{\rm i}\varphi(f)},\quad f\in L^{2}_{0},$ (2.12)

is

$\omega_{{\rm R},\beta}\big{(}W(f)\big{)}=e^{-\tfrac{1}{4}\langle
f,\coth(\beta|k|/2)f\rangle}.$ (2.13)

It is very convenient to use a purification of $\omega_{{\rm R},\beta}$, that
is, to represent this state as a vector state in a new Hilbert space
${\mathcal{H}}_{\rm R}$ (different from ${\mathcal{F}}$, (2.4)), in which the
operators $a^{\sharp}(f)$ (where $a^{\sharp}=a$ or $a^{\sharp}=a^{*}$) and
$W(f)$ are represented by operators

$a^{\sharp}(f)\mapsto\pi_{\rm R}\big{(}a^{\sharp}(f)\big{)},\qquad
W(f)\mapsto\pi_{\rm R}(W(f)).$ (2.14)

Here, $\pi_{\rm R}$ is a linear map, sending operators on $\mathcal{F}$ to
operators on ${\mathcal{H}}_{\rm R}$ and satisfying $\pi_{\rm
R}(X^{*})=\pi_{\rm R}(X)^{*}$ and $\pi_{\rm R}(XY)=\pi_{\rm R}(X)\pi_{\rm
R}(Y)$, i.e., $\pi_{\rm R}$ is a $*$ algebra representation. Let $O_{\rm R}$
be an arbitrary finite sum of products of creation and annihilation operators
and Weyl operators acting on $\mathcal{F}$. Then we have

$\omega_{{\rm R},\beta}(O_{\rm R})=\langle\Omega_{\rm R},\pi_{\rm R}(O_{\rm
R})\Omega_{\rm R}\rangle_{{\mathcal{H}}_{\rm R}}$ (2.15)

for a normalized vector $\Omega_{\rm R}\in{\mathcal{H}}_{\rm R}$. In this
sense, $\omega_{{\rm R},\beta}$ is represented by the vector $\Omega_{\rm R}$.
The explicit form of ${\mathcal{H}}_{\rm R}$, $\pi_{\rm R}$ and $\Omega_{\rm
R}$ is well known (Araki-Woods representation [4]), we give them in Section
3.1.

The purification of the uncoupled joint equilibrium state $\rho_{{\rm
S},\beta}\otimes\omega_{{\rm R},\beta}$ is obtained by taking the tensor
product: For operators $O$ which are are arbitrary finite sums of products of
bounded operators on ${\mathbb{C}}^{N}$, creation and annihilation operators
and Weyl operators (with test functions in $L^{2}_{0}$), we have

$\rho_{{\rm S},\beta}\otimes\omega_{{\rm R},\beta}(O)=\langle\Omega_{{\rm
S}{\rm R},\beta,0},\pi(O)\Omega_{{\rm S}{\rm
R},\beta,0}\rangle_{{\mathcal{H}}_{{\rm GNS}}},$ (2.16)

where (GNS stands for Gelfand-Naimark-Segal)

${\mathcal{H}}_{{\rm
GNS}}={\mathbb{C}}^{N}\otimes{\mathbb{C}}^{N}\otimes{\mathcal{H}}_{\rm
R},\quad\pi=\pi_{\rm S}\otimes\pi_{\rm R}\quad\mbox{and}\quad\Omega_{{\rm
S}{\rm R},\beta,0}=\Omega_{\rm S}\otimes\Omega_{\rm R}.$ (2.17)

###  2.2 Observable algebra $\mathcal{O}$ and correlation algebra
$\mathfrak{C}$

We introduce two spaces of single reservoir particle states (‘test
functions’).

  * (a)

The space of observable test functions is $L^{2}_{\rm obs}\subset
L^{2}({\mathbb{R}}^{3},d^{3}k)$, consisting of all functions of the form

$f(k)=\frac{|k|^{p}}{1+|k|^{p}}\big{(}1+|k|^{2}\big{)}\,h(k),$ (2.18)

where $p=-\tfrac{1}{2},\tfrac{1}{2}$ or $p>1$, and $h(k)$ is a function s.t.
for $j=0,1,2,3$, the derivatives $\tfrac{\partial^{j}}{\partial|k|^{j}}h(k)$
exist and are bounded for $|k|>0$, and satisfy $\lim_{|k|\rightarrow
0}\tfrac{\partial^{j}}{\partial|k|^{j}}h(k)\neq 0$.

  * (b)

The test function space for correlation operators $L^{2}_{\rm cor}$ consists
of all functions $f\in L^{2}_{\rm obs}$ such that $e^{\beta|k|}f\in
L^{2}({\mathbb{R}}^{3},d^{3}k)$, where $\beta$ is the inverse temperature.

In short, the functions in $L_{\rm obs}^{2}$ are three times differentiable
and have the infrared behaviour $f(k)\sim|k|^{p}$ (for $|k|$ small) and
ultraviolet behaviour $f(k)\sim|k|^{-q}$ (for $|k|$ large), for some $q>3$.
Functions in $L^{2}_{\rm cor}$ have to decay exponentially in the ultraviolet
region. The observable algebra is defined as

${\mathcal{O}}={\mathcal{B}}({\mathbb{C}}^{N})\otimes{\mathfrak{W}}_{0}(L^{2}_{\rm
obs})$ (2.19)

where ${\mathcal{B}}({\mathbb{C}}^{N})$ is the algebra of linear operators on
${\mathbb{C}}^{N}$ and where ${\mathfrak{W}}_{0}(L^{2}_{\rm obs})$ is the
algebra consisting of all finite sums and products of Weyl operators $W(f)$
with $f\in L^{2}_{\rm obs}$.222It is not hard to extend our results to the
case when $\mathcal{O}$ is the norm closure of (2.19) (i.e., for
${\mathfrak{W}}_{0}(L^{2}_{\rm obs})$ replaced by the $C^{*}$-algebra
${\mathfrak{W}}(L^{2}_{\rm obs})$), see the discussion after Theorem 2.2.
Next, introduce the polynomial algebra

${\mathcal{P}}={\rm Span}\big{\\{}B\otimes a^{\sharp}(f_{1})\cdots
a^{\sharp}(f_{n})\ :\ B\in{\mathcal{B}}({\mathbb{C}}^{N}),\ n\in{\mathbb{N}},\
f_{j}\in L^{2}_{\rm cor}\big{\\}},$ (2.20)

where ${\rm Span}$ is the linear span (finitely many complex linear
combinations) and $a^{\sharp}$ denotes either of $a$ or $a^{*}$, individually
in each factor in the product in (2.20). The set $\mathcal{P}$ consists of all
polynomials of creation and annihilation operators with coefficients in
${\mathcal{B}}({\mathbb{C}}^{N})$. Furthermore, define the set of operators

${\mathcal{X}}={\rm Span}\big{\\{}e^{{\rm i}\sum_{r=1}^{R}B_{r}\otimes
a^{\sharp}(f_{r})}\ :\ R\in{\mathbb{N}},\
B_{r}\in{\mathcal{B}}({\mathbb{C}}^{N}),\ f_{r}\in L^{2}_{\rm cor}\big{\\}}.$
(2.21)

The correlation algebra is defined as

${\mathfrak{C}}={\rm Span}\big{\\{}K_{1}\cdots K_{n}\ :\ n\in{\mathbb{N}},\
K_{j}\in{\mathcal{X}}\cup{\mathcal{P}}\big{\\}}.$ (2.22)

While the observable algebra ${\mathcal{O}}$ consist of bounded operators on
${\mathcal{H}}_{{\rm S}{\rm R}}$, the elements $K_{j}$ in (2.22) are generally
unbounded operators. One must be careful about the definition of the product
of such operators, showing up in (2.22). See Proposition 2.1.

###  2.3 Initial states

We use the notation $\rho(O)={\rm tr}(\rho O)$ for a density matrix $\rho$ and
an observable $O$. The initial states are of the form

$\rho_{{\rm S}{\rm R}}=\sum_{\alpha=1}^{\nu}\rho_{{\rm
S},\beta}\otimes\omega_{{\rm R},\beta}\big{(}K^{*}_{\alpha}\cdot
K_{\alpha}\big{)},$ (2.23)

where $\nu\in{\mathbb{N}}$ and the Kraus operators $K_{\alpha}$ belong to an
algebra of (possibly unbounded) operators ${\mathfrak{C}}$ acting on
${\mathcal{H}}_{{\rm S}{\rm R}}$. One may extend the results to $\nu=\infty$,
see the remarks after Theorem 2.2. The definition (2.23) gives $\rho_{{\rm
S}{\rm R}}$ as a state in the algebraic sense, as a functional on observables
of ${\rm S}{\rm R}$ observables $O\in\mathcal{O}$,

$\rho_{{\rm S}{\rm R}}(O)=\sum_{\alpha=1}^{\nu}\rho_{{\rm
S},\beta}\otimes\omega_{{\rm
R},\beta}\big{(}K^{*}_{\alpha}OK_{\alpha}\big{)}.$ (2.24)

The Kraus operators satisfy

$\sum_{\alpha=1}^{\nu}K^{*}_{\alpha}K_{\alpha}=\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul},$
(2.25)

so that $\rho_{{\rm S}{\rm R}}(\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul})=1$ is properly
normalized. The initial reduced system state is defined by the relation

$\rho_{\rm S}(O_{\rm S})=\rho_{{\rm S}{\rm R}}\big{(}O_{\rm
S}\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm R}\big{)},\quad\forall O_{\rm
S}\in{\mathcal{B}}({\mathbb{C}}^{N}).$ (2.26)

The state at time $t$ is given by

$\rho_{{\rm S}{\rm R}}(e^{{\rm i}tH_{\lambda}}\cdot e^{-{\rm
i}tH_{\lambda}})=\sum_{\alpha=1}^{\nu}\rho_{{\rm S},\beta}\otimes\omega_{{\rm
R},\beta}\big{(}K^{*}_{\alpha}\,e^{{\rm i}tH_{\lambda}}\cdot e^{-{\rm
i}tH_{\lambda}}K_{\alpha}\big{)}.$ (2.27)

Technically, it is defined in a standard way as follows (see Section 3.2). Use
the Dyson series to define the dynamics $e^{{\rm i}tH_{\lambda}}\cdot e^{-{\rm
i}tH_{\lambda}}$ starting from the uncoupled dynamics $e^{{\rm i}tH_{0}}\cdot
e^{-{\rm i}tH_{0}}$. Then apply the representation map $\pi$ (2.17) to the
Dyson series. One shows that the resulting series converges, and moreover,
that the limit equals $e^{{\rm i}tL_{\lambda}}\pi(\cdot)e^{-{\rm
i}tL_{\lambda}}$, where $L_{\lambda}$ is a self-adjoint operator on
${\mathcal{H}}_{{\rm GNS}}$, called the Liouville operator. The term in the
sum (2.27) is defined as

$\rho_{{\rm S},\beta}\otimes\omega_{{\rm
R},\beta}\big{(}K^{*}_{\alpha}\,e^{{\rm i}tH_{\lambda}}\cdot e^{-{\rm
i}tH_{\lambda}}K_{\alpha}\big{)}=\langle\Omega_{{\rm S}{\rm
R},\beta,0},\pi(K_{\alpha}^{*})e^{{\rm i}tL_{\lambda}}\pi(\cdot)e^{-{\rm
i}tL_{\lambda}}\pi(K_{\alpha})\Omega_{{\rm S}{\rm
R},\beta,0}\rangle_{{\mathcal{H}}_{{\rm GNS}}}.$ (2.28)

We prove the following result in Section 3.4:

###### Proposition 2.1

The functional $\rho_{{\rm S}{\rm R}}(e^{{\rm i}tH_{\lambda}}\cdot e^{-{\rm
i}tH_{\lambda}})$ given by (2.27) is a well defined state (positive, linear,
normalized functional) on $\mathcal{O}$, for any choice of the
$K_{j}\in{\mathfrak{C}}$.

One can show that $\rho_{{\rm S}{\rm R}}$ is also defined (finite) on
observables $O$ which are products of polynomials $B\otimes
a^{\sharp}(f_{1})\cdots a^{\sharp}(f_{n})$ and operators $e^{{\rm
i}\sum_{r=1}^{R}B_{r}\otimes a^{\sharp}(f_{r})}$ with test functions $f_{j}\in
L^{2}_{\rm obs}$. This is a larger class than ${\mathcal{O}}$. However, it
will be more difficult to give a proof of our main result, Theorem 2.2, for
unbounded observables. In the proof, we use the time uniform estimate
$\|e^{{\rm i}tH_{\lambda}}Oe^{-{\rm i}tH_{\lambda}}\|=\|O\|$, valid for
bounded observables $O$. For undbounded $O$ (say a product of creation and
annihilation operators) one has to find time uniform bounds of $e^{{\rm
i}tH_{\lambda}}Oe^{-{\rm i}tH_{\lambda}}$ in a weak sense (on suitable
functionals). This complicates the analysis and we do not address it here.

The collection of initial states does not depend the state $\rho_{{\rm
S},\beta}$ in the definition (2.23). Indeed, suppose we had put an arbitrary
system density matrix $\sigma_{\rm S}$ in the place of $\rho_{{\rm S},\beta}$
in (2.23). Then, since $\sigma_{\rm S}=K\rho_{{\rm S},\beta}K^{*}$ for
$K=\sqrt{\sigma_{\rm S}}\,(\rho_{{\rm S},\beta})^{-1/2}$, we obtain the same
class of initial states because changing the system reference state simply
amounts to a change in the Kraus operators. The same is not true for the role
of $\omega_{{\rm R},\beta}$ in (2.23). If we replace there $\omega_{{\rm
R},\beta}$ by $\omega_{{\rm R},\beta}(K^{*}\cdot K)$ for some reservoir
operator $K\in{\mathfrak{C}}$ (bounded or unbounded), then we still obtain the
same class of initial states. The set of such states $\omega_{{\rm
R},\beta}(K^{*}\cdot K)$ are called normal states w.r.t. $\omega_{{\rm
R},\beta}$, they form the folium of $\omega_{{\rm R},\beta}$. However, not all
states of the reservoir are of this form; for instance, $\omega_{{\rm
R},\beta^{\prime}}$ is not normal w.r.t. $\omega_{{\rm R},\beta}$ unless
$\beta=\beta^{\prime}$. Normal states differ from each other only
‘quasilocally’. We refer to [6, 19] for a more precise discussion of this
point.

Example.  An admissible initial state is for instance

$\rho_{{\rm S}{\rm R}}=\Lambda(\rho_{{\rm S},\beta}\otimes\omega_{{\rm
R},\beta})=\frac{1}{Z}\,e^{{\mathcal{E}}}\big{(}\rho_{{\rm
S},\beta}\otimes\omega_{{\rm R},\beta}\big{)}\,e^{{\mathcal{E}}^{*}},$

where $Z$ is a normalization factor and ${\mathcal{E}}=\sum_{j}B_{j}\otimes
a^{*}(f_{j})+\sum_{k}D_{k}\otimes a(f_{k})$ is an arbitrary expression linear
in the creation and annihilation operators. This is a state in which ${\rm S}$
and ${\rm R}$ are entangled, c.f. [39].

###  2.4 The main result

Assumptions.

  * (A1)

Smoothness of the form factor. We assume that the form factor $g(k)$ in the
interaction (2.5) satisfies

$g(k)=\frac{|k|^{p}}{1+|k|^{p+q}}h(k),$ (2.29)

for some values $p=-\tfrac{1}{2},\tfrac{1}{2},\tfrac{3}{2}$ or $p>2$ and
$q>2$, and where $h(k)$ is a function four times differentiable with respect
to $|k|$ for $|k|>0$, such that $h$ and its derivatives are bounded and
$\lim_{|k|\rightarrow 0}\partial^{j}_{|k|}h(k)\neq 0$, $j=0,\ldots,4$.

  * (A2)

    * (a)

Fermi Golden Rule Condition: The reservoir spectral density is defined as

$J(\omega)=\tfrac{1}{2}\pi\omega^{2}\int_{S^{2}}|g(\omega,\Sigma)|^{2}d\Sigma,\qquad\omega\geq
0,$ (2.30)

where $g(k)$ is the form factor in the interaction (2.5) and the integral is
over the (polar and azimuthal) angles. We assume that

$\langle\phi_{m},G\phi_{n}\rangle J(|E_{m}-E_{n}|)\neq 0\qquad\mbox{for all
system energies $E_{m}\neq E_{n}$},$ (2.31)

and where $\langle\phi_{m},G\phi_{n}\rangle$ are the matrix elements of the
coupling operator $G$ (2.5) in the system energy basis.

    * (b)

Simplicity of resonance energies. We assume that the so-called level shift
operators $\Lambda_{e}$, $e\in{\mathcal{E}}_{0}=\\{E_{m}-E_{n}\ :\
m,n=1,\ldots,N\\}$, have simple eigenvalues. The $\Lambda_{e}$ are explicit
matrices describing the second order ($\lambda^{2}$) corrections to the energy
differences $e\in{\mathcal{E}}_{0}$, given in Section 3.6.

The assumption (A1) on the form factor is more restrictive than $g\in
L^{2}_{\rm obs}$ (c.f. (2.18)) in that it requires a more stringent infrared
behaviour (values of $p$) and the existence of one more derivative, relative
to functions in $L^{2}_{\rm obs}$. The condition (2.31) ensures that the
interaction does not suppress second order ($\lambda^{2}$) transition
processes in S due to the coupling with R. The reservoir spectral density
(2.30) governs these transitions since the coupling (2.5) is linear in the
field operator. This linear form of the interaction is not necessary for our
method to work – other interactions will lead to explicit conditions of
effective coupling but are not expressed in terms of $J(\omega)$. The
assumptions (A1) together with (A2a) guarantee that the coupled system-
reservoir complex has a unique stationary state, the coupled equilibrium
state, for small nonzero $\lambda$, see [27]. (A2b) is a simplifying
assumption that can be quite easily removed by a slightly more cumbersome
analysis.

Here is our main result.

######  Theorem 2.2 (SR dynamics for correlated initial states)

There is a constant $\lambda_{0}>0$ such that if $|\lambda|<\lambda_{0}$, then
the following holds. Let $\rho_{{\rm S}{\rm R}}$ be an initial system-
reservoir state of the form (2.24) and let $\rho_{\rm S}$ be its reduction to
${\rm S}$ (2.26). Then for all $t\geq 0$, $O\in\mathcal{O}$,

$\rho_{{\rm S}{\rm R}}\big{(}e^{{\rm i}tH_{\lambda}}Oe^{-{\rm
i}tH_{\lambda}}\big{)}=\big{(}e^{t{\mathcal{L}}_{\rm S}(\lambda)}\rho_{\rm
S}\otimes\omega_{{\rm R},\beta}\big{)}(O)+\chi(\lambda,t,O)+R(\lambda,t,O),$
(2.32)

where ${\mathcal{L}}_{\rm S}(\lambda)$ is the Davies generator (1.2) and the
remainder $R(\lambda,t,O)$ satisfies

$|R(\lambda,t,O)|\leq C(O)|\lambda|^{1/4}.$ (2.33)

The dispersive term satisfies

$\displaystyle\chi(\lambda,t,O)$ $\displaystyle=$ $\displaystyle
0\qquad\mbox{if\ \ $\rho_{{\rm S}{\rm R}}=\rho_{\rm S}\otimes\omega_{{\rm
R},\beta}$}$ (2.34) $\displaystyle\chi(\lambda,t,O_{\rm
S}\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm R})$ $\displaystyle=$ $\displaystyle
0$ (2.35) $\displaystyle|\chi(\lambda,t,O)|$ $\displaystyle\leq$
$\displaystyle\frac{C(O)}{1+t^{3}},$ (2.36)

and moreover,

$|\chi(\lambda,t,O)-\big{(}\rho_{{\rm S}{\rm R}}-\rho_{\rm
S}\otimes\omega_{{\rm R},\beta}\big{)}(e^{{\rm i}tH_{0}}Oe^{-{\rm
i}tH_{0}})|\leq C(O)\,|\lambda|.$ (2.37)

The constants $C(O)$ depend on $O$ as well as the initial state $\rho_{{\rm
S}{\rm R}}$ (but are independent of $t,\lambda$). For $O=O_{\rm
S}\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm R}$ we have $C(O_{\rm
S}\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm R})\leq c\|O_{\rm S}\|$ with $c$
independent of $O_{\rm S}$.

###  2.5 Implications

Theorem 2.2 has the following consequences.

  * (1)

Validity of the Born approximation for initially factorized states

For initial states $\rho_{{\rm S}{\rm R}}=\rho_{\rm S}\otimes\omega_{{\rm
R},\beta}$, where $\rho_{\rm S}$ is any system density matrix, and for all
$t\geq 0$, $O\in\mathcal{O}$, we have

$\rho_{{\rm S}{\rm R}}\big{(}e^{{\rm i}tH_{\lambda}}Oe^{-{\rm
i}tH_{\lambda}}\big{)}=\big{(}e^{t{\mathcal{L}}_{\rm S}(\lambda)}\rho_{\rm
S}\otimes\omega_{{\rm R},\beta}\big{)}(O)+{\mathbf{O}}(|\lambda|^{1/4}).$
(2.38)

This shows that if the system and reservoir are initially in a product state
with the reservoir in equilibrium, then the total state stays (up to
$\mathbf{O}(|\lambda|^{1/4})$ of product form for all times, and the reservoir
stays in its equilibrium state.

  * (2)

Validity of the Markov approximation for initially correlated states

Taking in (2.32) observables of the form $O=O_{\rm S}\otimes\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}_{\rm R}$ shows the validity of the Markovian approximation,
(1.4), even for initially correlated ${\rm S}{\rm R}$ states. Namely, denoting
by $\rho_{\rm S}^{t}$ the reduction of the full state $\rho_{{\rm S}{\rm
R}}=\rho_{{\rm S}{\rm R}}(e^{{\rm i}tH_{\lambda}}\cdot e^{-{\rm
i}tH_{\lambda}})$ at time $t$, and $\rho_{\rm S}=\rho_{\rm S}^{t=0}$, we have

$\sup_{t\geq 0}\big{\|}\rho_{\rm S}^{t}-e^{t{\mathcal{L}}_{\rm
S}(\lambda)}\rho_{\rm S}\big{\|}\leq C|\lambda|^{1/4},$ (2.39)

for a constant $C$ independent of $\lambda$, but generally depending on the
initial state $\rho_{{\rm S}{\rm R}}$. (As $\rho_{\rm S}^{t}$ is a finite-
dimensional density matrix, it is not necessary to specify which norm we take
in (2.39). We may take the trace norm.)

  * (3)

Decay of correlations and emergence of the Born approximation regime

The term $\chi$ in (2.32) describes the evolution of the correlation between
the system and reservoir. According to (2.36) it decays as $t^{-3}$. At $t=0$,
(2.32) gives $\chi(\lambda,t,O)=\rho_{{\rm S}{\rm R}}(O)-{\rm tr}_{\rm
R}\\{\rho_{{\rm S}{\rm R}}\\}\otimes\omega_{{\rm
R},\beta}+{\mathbf{O}}(|\lambda|^{1/4})$, so if the initial state $\rho_{{\rm
S}{\rm R}}$ differs from ${\rm tr}\\{\rho_{{\rm S}{\rm
R}}\\}\otimes\omega_{{\rm R},\beta}$ then for small times, the correlation
term $\chi(\lambda,t,O)$ can dominate the Markovian term
$(e^{t{\mathcal{L}}_{\rm S}(\lambda)}\rho_{\rm S}\otimes\omega_{{\rm
R},\beta})(O)$ in (2.32). Whether and by how much this happens depends on the
observable $O$ – as explained in the previous point, for $O=O_{\rm
S}\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm R}$ we always have
$\chi(\lambda,t,O)=0$. To see which one of the Markovian or the correlation
contributions are dominant, we use the spectral decomposition [29]

$e^{t{\mathcal{L}}_{\rm S}(\lambda)}=\sum_{j}e^{{\rm
i}t(e_{j}+\lambda^{2}a_{j})}P_{j},$ (2.40)

where the $P_{j}$ are spectral projections, the $e_{j}\in\mathbb{R}$ are the
Bohr energies of the system (differences of eigenvalues of $H_{\rm S}$) and
$a_{j}\in\mathbb{C}$, ${\rm Im}a_{j}\geq 0$ are the complex energy corrections
(resonances) induced by the interaction with the reservoir. The Markovian
dynamics exhibits exponential time decay $\sim e^{-\lambda^{2}\gamma t}$ in
all directions $P_{j}$ but one, which represents the projection onto the
system equilibrium state at temperature $\beta$ (and for which
$e_{j}=a_{j}=0$). The decay (approach to equilibrium) is exponential and very
slow for small coupling $\lambda$. On the other hand, the polynomial decay of
the correlation term, $t^{-3}$, happens at a rate which is independent of
$\lambda$, meaning that the constant in (2.36) does not depend on $\lambda$.
(Indeed, that constant encodes the fact that the reservoir dynamics alone is
dispersive away from the projection onto the reservoir equilibrium state.) It
follows that for small coupling, the correlation term $\chi$ can be dominant
for small times, but as time increases, the Markovian term becomes dominant.
Then a long time later ($\lambda$ small), the exponentially decaying functions
lie below the power decay function, and the correlation term is leading once
again, even though by that time, both those terms are smaller than the
remainder $\mathbf{O}(|\lambda|^{4})$.

Further remarks.

  * (a)

The power of the decay in (2.36) depends on the smoothness of the form factor
$g$, (2.5), which in turn determines the decay speed of the reservoir
correlation function [30, 31]. We get higher powers $\sim t^{-n}$ assuming
that $g$ is $(n+1)$ times differentiable and that the test functions in
$L^{2}_{\rm obs}$ are $n$ times differentiable. Under a certain analyticity
condition on $g$ (see [29]) and on the test functions in $L^{2}_{\rm obs}$,
one can show that the time decay in (2.36) is exponential, with a
$\lambda$-independent decay rate.

  * (b)

We cannot ascertain in general that the constant $C(O)$ in Theorem 2.2 are
bounded in the values of $t>0$ for time-dependent observables of the form
$O=e^{{\rm i}tH_{0}}O^{\prime}e^{-{\rm i}tH_{0}}$. This is so since the
constant will depend on ‘Sobolev norms’ of test functions appearing in the
observables, that is, on their smoothness (the
$L^{2}({\mathbb{R}}^{3},d^{3}k)$ norm of their derivatives w.r.t. $|k|$). But
$e^{{\rm i}tH_{\rm R}}W(f)e^{-{\rm i}tH_{\rm R}}=W(e^{{\rm i}|k|t}f)$ has a
test function $e^{{\rm i}|k|t}f(k)$ that is rough (large $|k|$-derivative) for
large $t$. Nevertheless, if $O=O_{\rm S}\otimes\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}_{\rm R}$, then $C(e^{{\rm i}tH_{\rm S}}O_{\rm S}e^{-{\rm
i}H_{\rm S}t}\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm R})\leq c\|O_{\rm S}\|$ is bounded
uniformly in time.

  * (c)

In the weak coupling regime one takes simultaneously $\lambda\rightarrow 0$
and $t\rightarrow\infty$, keeping $\tau=\lambda^{2}t\in\mathbb{R}$ fixed. By
(2.33), (2.36) and the previous discussion point,

$\displaystyle\lim_{\lambda\rightarrow
0}R\Big{(}\lambda,\tfrac{\tau}{\lambda^{2}},(e^{-{\rm
i}\frac{\tau}{\lambda^{2}}H_{\rm S}}\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm R})O(e^{{\rm
i}\frac{\tau}{\lambda^{2}}H_{\rm S}}\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm R})\Big{)}$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle\lim_{\lambda\rightarrow
0}(\rho_{{\rm S}{\rm R}}-\rho_{\rm S}\otimes\omega_{{\rm
R},\beta})\Big{(}e^{{\rm i}\frac{\tau}{\lambda^{2}}H_{0}}(e^{-{\rm
i}\frac{\tau}{\lambda^{2}}H_{\rm S}}\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm R})O(e^{{\rm
i}\frac{\tau}{\lambda^{2}}H_{\rm S}}\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm R})e^{-{\rm
i}\frac{\tau}{\lambda^{2}}H_{0}}\Big{)}$ $\displaystyle=$ $\displaystyle 0.$

Thus (2.32) gives for all $\tau>0$, $O\in\mathcal{O}$

$\lim_{\lambda\rightarrow 0}\ \rho_{{\rm S}{\rm R}}\Big{(}e^{{\rm
i}\frac{\tau}{\lambda^{2}}H_{\lambda}}(e^{-{\rm
i}\frac{\tau}{\lambda^{2}}H_{\rm S}}\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm R})O(e^{{\rm
i}\frac{\tau}{\lambda^{2}}H_{\rm S}}\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm R})e^{-{\rm
i}\frac{\tau}{\lambda^{2}}H_{\lambda}}\Big{)}=\big{(}e^{\tau\mathcal{K}}\rho_{\rm
S}\otimes\omega_{{\rm R},\beta}\big{)}(O).$

This shows that in the weak coupling scaling limit, the total system-reservoir
state (in the interaction picture when the system dynamics is removed) is of
the product form $(e^{\tau\mathcal{K}}\rho_{\rm S})\otimes\omega_{{\rm
R},\beta}$ for all $\tau>0$. This fact was already observed in [38].

  * (d)

If $O$ belongs to the norm closure of ${\mathcal{O}}$, then take an arbitrary
$\epsilon>0$ and $O_{\epsilon}\in\mathcal{O}$ such that
$\|O-O_{\epsilon}\|<\epsilon$. The quantity $\rho_{{\rm S}{\rm R}}(e^{{\rm
i}tH_{\lambda}}Oe^{-{\rm i}tH_{\lambda}})$ is still well defined and we have
$\rho_{{\rm S}{\rm R}}(e^{{\rm i}tH_{\lambda}}Oe^{-{\rm
i}tH_{\lambda}})-\rho_{{\rm S}{\rm R}}(e^{{\rm
i}tH_{\lambda}}O_{\epsilon}e^{-{\rm i}tH_{\lambda}})=O(\epsilon)$, uniformly
in $t$. Theorem (2.2) then gives the expansion (2.32) for $\rho_{{\rm S}{\rm
R}}(e^{{\rm i}tH_{\lambda}}O_{\epsilon}e^{-{\rm i}tH_{\lambda}})$, with
constants $C(O_{\epsilon})$ depending on $\epsilon$. In the main term,
$\big{(}e^{t{\mathcal{L}}_{\rm S}(\lambda)}\rho_{\rm S}\otimes\omega_{{\rm
R},\beta}\big{)}(O_{\epsilon})+\chi(\lambda,t,O_{\epsilon})$, we can replace
$O_{\epsilon}$ again by $O$, making an error of $O(\epsilon)$, uniformly in
$\lambda$ and $t$ (see also (3.49)). This shows that (2.32) remains valid for
any $O$ in the norm closure of $\mathcal{O}$, modulo adding an arbitrarily
small term $O(\epsilon)$ and allowing the constants $C(O)$ to depend also on
$\epsilon$. One may use a similar argument to extend the result of Theorem 2.2
to $\nu=\infty$ in (2.23).

Connections. The mathematical method we are using is based on [30, 31]. To our
knowledge, this is the only approach able to handle perturbation theory in
small $\lambda$ valid for all times $t\geq 0$. On the physics side, our work
is close to [38, 39], where the authors show that the master equation is valid
for correlated initial states in the weak coupling limit (see the Remark (c)
above). Their technique is based on the Nakajima-Zwanzig projection method and
does not extend beyond the weak coupling scaling regime. In some aspects, the
setup in [38, 39] is more general (allowing to treat NESS for example – an
extension of our methods to include this is planned) and in some aspects it is
less general (their Kraus operators have to be bounded for the main result in
[38] – even though unbounded ones are used in applications given in [39]).

One of our main conclusions is that initial correlations do not invalidate the
Markovian approximation. This means that the distinguishability (trace
distance) of reduced system states cannot increase during the evolution (by
more than $\mathbf{O}(|\lambda|^{1/4})$). An important feature responsible for
this is that the reservoir dynamics is dispersive, meaning that the evolution
converges to the stationary (equilibrium) state. This is built into the model
by taking the infinite volume limit (continuous reservoir mode frequencies)
and by the continuity of the spectrum of $H_{\rm R}$ (see also [30, 31, 29]).
In contrast, if ${\rm R}$ is finite, an $M$-level system with discrete energy
levels, then different effects appear. Those are studied in a variety of
papers. It is shown in [25, 13] that for finite-dimensional ${\rm S}$ and
${\rm R}$, the backflow of information ${\rm R}\rightarrow{\rm S}$ can lead to
an increase in the distinguishability if the ${\rm S}{\rm R}$ state is
initially correlated, an effect which cannot happen for initial product
states. The increase is a measure for non-Markovianity of the dynamics. In
this situation, initial correlations lead to non-Markovianity of the system
dynamics. A related question is whether and how the system dynamics can be
represented by completely positive, Kraus representation maps. In [20], the
authors consider finite-dimensional ${\rm S}$ and ${\rm R}$ and show that a
Kraus representation is valid for all times and for general initial
correlations if and only if the joint dynamics is local unitary (no ${\rm
S}{\rm R}$ coupling). It is shown in [34] that the dynamics of an open
$N$-level system can be described by at most $N^{2}$ completely positive trace
preserving maps. The result is built on a decomposition of the (correlated)
initial state using a so-called bath-positive decomposition.

In [3], a new correlation picture approach is used to show that the reduced
system dynamics satisfies a Lindblad-like master equation, in which the jump
operators depend on the initial ${\rm S}{\rm R}$ correlation, yielding a
nonlinear evolution equation. The philosophy there is to build a weak
correlation, rather than a weak coupling, perturbation theory. The
corresponding perturbation series does not converge uniformly in time, but
allows to analyze a strongly interacting ${\rm S}{\rm R}$ dynamics for finite
times.

##  3 Proof of Theorem 2.2

###  3.1 Purification of $\rho_{{\rm S}{\rm R}}$

Our approach follows the works [30, 31] (see also [23]) and we adopt the
notation of [31]. The first step is to work with a purification of the initial
state $\rho_{{\rm S}{\rm R}}$. Consider $\rho_{\rm S}\otimes\omega_{{\rm
R},\beta}$, where $\rho_{\rm S}$ is an arbitrary density matrix acting on
${\mathbb{C}}^{N}$ and $\omega_{{\rm R},\beta}$ is the reservoir equilibrium
state (2.13). The state $\rho_{\rm S}\otimes\omega_{{\rm R},\beta}$ is
represented by a vector

$\Psi_{0}=\Psi_{\rm S}\otimes\Omega_{\rm R}\in{\mathcal{H}}_{\rm GNS}$ (3.1)

in the new ‘purification’ Hilbert space

${\mathcal{H}}_{\rm GNS}={\mathcal{H}}_{\rm S}\otimes{\mathcal{H}}_{\rm R}$
(3.2)

where

${\mathcal{H}}_{\rm
S}={\mathbb{C}}^{N}\otimes{\mathbb{C}}^{N}\quad\mbox{and}\quad{\mathcal{H}}_{\rm
R}=\bigoplus_{n\geq 0}L^{2}_{\rm sym}\big{(}({\mathbb{R}}\times
S^{2})^{n},(du\times d\Sigma)^{n}\big{)}$ (3.3)

is the (symmetric) Fock space over the one-particle space

$L^{2}({\mathbb{R}}\times S^{2},du\times d\Sigma)\equiv
L^{2}({\mathbb{R}}\times S^{2}).$ (3.4)

Here, $d\Sigma$ is the uniform measure over the unit sphere
$S^{2}\subset{\mathbb{R}}^{3}$. The link between $\rho_{\rm
S}\otimes\omega_{{\rm R},\beta}$ and (3.1) is given by a representation map

$\pi=\pi_{\rm S}\otimes\pi_{\rm
R}:{\mathcal{O}}\rightarrow{\mathcal{L}}({\mathcal{H}}),$ (3.5)

mapping observables $O$ acting on ${\mathbb{C}}^{N}\otimes{\mathcal{F}}$
(before purification (2.2)) to operators on ${\mathcal{H}}_{\rm GNS}$ (after
purification (3.2)), in such a way that

$\big{(}\rho_{\rm S}\otimes\omega_{{\rm
R},\beta}\big{)}(O)=\langle\Psi_{0},\pi(O)\Psi_{0}\rangle_{{\mathcal{H}}_{\rm
GNS}}.$ (3.6)

The vector $\Omega_{\rm R}$ in (3.1) is the vacuum vector in the Fock space
${\mathcal{H}}_{\rm R}$ (3.3). It is the purification of the reservoir thermal
equilibrium. The vector $\Psi_{\rm
S}\in{\mathbb{C}}^{N}\otimes{\mathbb{C}}^{N}$ is the purification of the
system density matrix $\rho_{\rm S}$. For example, the purification of the
system Gibbs state $\rho_{{\rm S},\beta}$ (2.8) is given by

$\Omega_{{\rm S},\beta}=Z_{{\rm S},\beta}^{-1/2}\sum_{j=1}^{N}e^{-\beta
E_{j}/2}\phi_{j}\otimes\phi_{j}\in{\mathbb{C}}^{N}\otimes{\mathbb{C}}^{N},$
(3.7)

where $Z_{{\rm S},\beta}={\rm tr}_{\rm S}\,e^{-\beta H_{\rm S}}$. The
representation map $\pi$, (3.5) is explicitly given by

$\displaystyle\pi_{\rm S}(A)$ $\displaystyle=$ $\displaystyle
A\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm S},$ (3.8) $\displaystyle\pi_{\rm
R}(W(f))$ $\displaystyle=$ $\displaystyle W_{\beta}(\tau_{\beta}f),$ (3.9)

where

$\tau_{\beta}:L^{2}({\mathbb{R}}^{3},d^{3}k)\rightarrow
L^{2}({\mathbb{R}}\times S^{2},du\times d\Sigma)$ (3.10)

takes a function $f(k)$, $k\in{\mathbb{R}}^{3}$, into the function
$(\tau_{\beta}f)(u,\Sigma)$, $u\in\mathbb{R}$, $\Sigma\in S^{2}$, defined by

$\big{(}\tau_{\beta}f\big{)}(u,\Sigma)=\sqrt{\frac{u}{1-\,{\rm e}^{-\beta
u}}}\ |u|^{1/2}\left\\{\begin{array}[]{ll}f(u,\Sigma),&u\geq 0\\\
-\bar{f}(-u,\Sigma)&u<0\end{array}\right..$ (3.11)

On the right side of (3.11), $g$ is represented in spherical coordinates,
$u=|k|\geq 0$, $\Sigma\in S^{2}$. The Weyl operator $W(f)$ on the left side of
(3.9) is given by (2.12) and is defined for $f\in
L^{2}({\mathbb{R}}^{3},d^{3}k)$. The Weyl operator on the right side is
defined for functions $f\in L^{2}({\mathbb{R}}\times S^{2},du\times d\Sigma)$,
as $W_{\beta}(f)=e^{{\rm i}\varphi_{\beta}(f)}$, where
$\varphi_{\beta}(f)=\frac{1}{\sqrt{2}}[a_{\beta}^{*}(f)+a_{\beta}(f)]$ is the
field operator in the Fock space ${\mathcal{H}}_{\rm R}$, (3.3), smoothed out
with $f$. The representation map $\pi_{\rm R}$ is also defined on creation
operators, by the explicit formula

$\pi_{\beta}(a^{*}(f))=a_{\beta}^{*}\Big{(}\textstyle\sqrt{\frac{u}{1-\,{\rm
e}^{-\beta u}}}\
|u|^{1/2}f(u,\Sigma)\chi_{+}(u)\Big{)}-a_{\beta}\Big{(}\textstyle\sqrt{\frac{u}{1-\,{\rm
e}^{-\beta u}}}\ |u|^{1/2}\bar{f}(-u,\Sigma)\chi_{-}(u)\Big{)},$ (3.12)

where $\chi_{+}(u)=1$ if $u\geq 0$ and $\chi_{+}(u)=0$ for $u<0$ (and
$\chi_{-}(u)=1-\chi_{+}(u)$). We refer to [6, 29, 31, 32] for more detail. The
purification of the initial state $\rho_{{\rm S}{\rm R}}$ (2.24) is given by

$\sum_{\alpha=1}^{\nu}\rho_{{\rm S},\beta}\otimes\omega_{{\rm
R},\beta}\big{(}K_{\alpha}^{*}OK_{\alpha}\big{)}=\sum_{\alpha=1}^{\nu}\langle\Omega_{{\rm
S}{\rm R},\beta,0},\pi(K_{\alpha}^{*})\pi({O})\pi(K_{\alpha})\Omega_{{\rm
S}{\rm R},\beta,0}\rangle,$ (3.13)

where the inner product is that of ${\mathcal{H}}_{\rm GNS}$ (3.2) and

$\Omega_{{\rm S}{\rm R},\beta,0}=\Omega_{{\rm S},\beta}\otimes\Omega_{\rm R}$
(3.14)

is the uncoupled equilibrium (KMS) state. We note that $\pi$ is well defined
on polynomials in creation and annihilation operators (and limits thereof),
see (2.11), and $\pi({\mathfrak{C}})$ is a well defined set of unbounded
operators on ${\mathcal{H}}_{\rm GNS}$. We prove in Lemma 3.4 below that
$\pi(K)\Omega_{{\rm S}{\rm R},\beta,0}$ is well defined for any
$K\in\mathfrak{C}$. This guarantees that (3.13) is well defined. One can see
the right hand side of (3.13) as the definition of our state $\rho_{{\rm
S}{\rm R}}$.

###  3.2 Coupled dynamics and Liouville operator

The uncoupled Heisenberg dynamics $\alpha^{t}_{0}(O)=e^{{\rm
i}tH_{0}}Oe^{-{\rm i}tH_{0}}$ leaves $\mathcal{O}$ invariant (see (2.19)), but
the interacting dynamics $e^{{\rm i}tH_{\lambda}}Oe^{-{\rm i}tH_{\lambda}}$
does not. It is thus not a priori clear how to represent $e^{{\rm
i}tH_{\lambda}}Oe^{-{\rm i}tH_{\lambda}}$ as an operator in
${\mathcal{H}}_{\rm GNS}$. However, we can extend the domain of $\pi$ and
define a self-adjoint Liouville operator on $\mathcal{H}_{\rm GNS}$, such that
for $O\in\mathcal{O}$,

$\pi\big{(}e^{{\rm i}tH_{\lambda}}{O}e^{-{\rm i}tH_{\lambda}}\big{)}=e^{{\rm
i}tL_{\lambda}}\pi({O})e^{-{\rm i}tL_{\lambda}}.$ (3.15)

To do this we proceed in a standard way using the Dyson expansion

$e^{{\rm i}tH_{\lambda}}Oe^{-{\rm
i}tH_{\lambda}}=\alpha^{t}_{0}(O)+\sum_{n\geq 1}({\rm
i}\lambda)^{n}\int_{0\leq t_{1}\leq\cdots\leq t_{n}\leq t}dt_{1}\cdots dt_{n}\
T_{t_{1},\ldots,t_{n}}(O),$ (3.16)

where

$T_{t_{1},\ldots,t_{n}}(O)=[\alpha^{t_{n}}_{0}(V),[\alpha_{0}^{t_{n-1}}(V),\ldots,[\alpha^{t_{1}}_{0}(V),\alpha^{t}_{0}(O)]\ldots]]$
(3.17)

is the multiple commutator and $V=G\otimes\varphi(g)$ is the interaction
operator (2.5). The series (3.16) converges in the strong sense on suitable
states. It is easy to check that

$\pi(e^{{\rm i}tH_{0}}Oe^{-{\rm i}tH_{0}})=e^{{\rm i}tL_{0}}\pi(O)e^{-{\rm
i}tL_{0}},$ (3.18)

where the uncoupled Liouville operator is (not writing obvious factors
$\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}$)

$L_{0}=L_{\rm S}+L_{\rm R},\quad L_{\rm S}=H_{\rm S}\otimes\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}_{\rm S}\ -\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm S}\otimes H_{\rm
S},\quad L_{\rm R}=d\Gamma(u).$ (3.19)

Here, $d\Gamma(u)$ is the second quantization of multiplication by the radial
variable $u\in{\mathbb{R}}$, acting on the Fock space ${\mathcal{H}}_{\rm R}$
(3.3). In particular, $\,{\rm e}^{{\rm i}tL_{\rm R}}a_{\beta}(f)e^{-{\rm
i}tL_{\rm R}}=a_{\beta}(e^{{\rm i}tu}f)$ is a Bogoliubov transformation. We
now apply $\pi$ to the right side of (3.16). More precisely, one shows that
$\pi$ applied to the truncated series $\sum_{n=1}^{N}\ldots$ has a strong
limit as $N\rightarrow\infty$. Due to the structure of the sum, the limit
operator has again the form of a Dyson series and equals $e^{{\rm
i}tL_{\lambda}}\pi(O)e^{-{\rm i}tL_{\lambda}}$ (see for instance Section 2.1.3
of [18] for details). The self-adjoint operator $L_{\lambda}$ is called the
Liouville operator and has the form

$L_{\lambda}=L_{0}+\lambda I,$ (3.20)

where the interaction operator $I$ in (3.20) is given by
$I=\pi\big{(}G\otimes\varphi(g)\big{)}-J\pi\big{(}G\otimes\varphi(g)\big{)}J$
(compare with (2.5)), where $J$ is the modular conjugation. The explicit form
of $J$ is well known, see e.g. (1.21) of [22]. One then gets the expression
[22, 29]

$I=G\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm
S}\otimes\varphi_{\beta}(\tau_{\beta}g)-\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm
S}\otimes{\mathcal{C}}G{\mathcal{C}}\otimes\varphi_{\beta}(e^{-\beta
u/2}\tau_{\beta}g),$ (3.21)

where $\mathcal{C}$ is the operator taking complex conjugation of components
of vectors in ${\mathbb{C}}^{N}$ written in the eigenbasis of $H_{\rm S}$. The
construction of $L_{\lambda}$ associated to $H_{\lambda}$ is well understood
[21, 5, 12, 29]. The upshot is that the right side of (3.15) defines the
dynamics of the infinitely extended system.

From perturbation theory of equilibrium (KMS) states [12, 6] we know that the
interacting system-reservoir complex has a unique equilibrium state,
represented by a normalized vector $\Omega_{{\rm S}{\rm
R},\beta,\lambda}\in{\mathcal{H}}$, satisfying

$L_{\lambda}\Omega_{{\rm S}{\rm R},\beta,\lambda}=0.$ (3.22)

The vector $\Omega_{{\rm S}{\rm R},\beta,\lambda}$ is analytic in $\lambda$ at
the origin, with (c.f. (3.14))

$\|\Omega_{{\rm S}{\rm R},\beta,\lambda}-\Omega_{{\rm S}{\rm R},\beta,0}\|\leq
C|\lambda|.$ (3.23)

###  3.3 Resonance expansion and proof of Theorem 2.2

The expectation of a system-reservoir observable $O\in\mathcal{O}$ in the
state $\rho_{{\rm S}{\rm R}}$ at time $t$ is

$\rho_{{\rm S}{\rm R}}(e^{{\rm i}tH_{\lambda}}{O}e^{-{\rm
i}tH_{\lambda}})=\sum_{\alpha=1}^{\nu}\rho_{{\rm S},\beta}\otimes\omega_{{\rm
R},\beta}\big{(}K^{*}_{\alpha}e^{{\rm i}tH_{\lambda}}{O}e^{-{\rm
i}tH_{\lambda}}K_{\alpha}\big{)}.$ (3.24)

We now analyze one of the terms in the sum, for a fixed $\alpha$, writing
simply $K$ for $K_{\alpha}$, and we will restore the sum over the $\alpha$ at
the end. According to (3.13), (3.15) we have

$\rho_{{\rm S},\beta}\otimes\omega_{{\rm R},\beta}\big{(}K^{*}\,{\rm e}^{{\rm
i}tH_{\lambda}}{O}\,{\rm e}^{-{\rm i}tH_{\lambda}}K\big{)}=\langle\Omega_{{\rm
S}{\rm R},\beta,0},\pi(K^{*})e^{{\rm i}tL_{\lambda}}\pi({O})e^{-{\rm
i}tL_{\lambda}}\pi(K)\Omega_{{\rm S}{\rm R},\beta,0}\rangle,$ (3.25)

where $\Omega_{{\rm S}{\rm R},\beta,0}$ is the uncoupled KMS state (3.14). The
representation $\pi^{\prime}(\cdot)=J\pi(\cdot)J$ has the property that
$\pi^{\prime}(X)$ and $\pi(Y)$ commute for all operators $X,Y$ (Tomita-
Takesaki, [6, 29]); for unbounded $X,Y$ the commutation is understood in the
strong (or weak) sense on suitable vectors. We now use the identity

$\pi(X)\Omega_{{\rm S}{\rm R},\beta,0}=\pi^{\prime}\big{(}e^{-\beta
H_{0}/2}X^{*}e^{\beta H_{0}/2}\big{)}\Omega_{{\rm S}{\rm R},\beta,0},$ (3.26)

which holds provided the operator $e^{-\beta H_{0}/2}X^{*}e^{\beta H_{0}/2}$
is well defined (note that $e^{\beta H_{0}/2}$ is unbounded). The relation
(3.26) is derived as follows. We have

$\displaystyle\pi(X)\Omega_{{\rm S}{\rm R},\beta,0}$ $\displaystyle=$
$\displaystyle J\Delta^{1/2}\pi(X^{*})J\Delta^{1/2}\Omega_{{\rm S}{\rm
R},\beta,0}$ (3.27) $\displaystyle=$ $\displaystyle
J\Delta^{1/2}\pi(X^{*})\Delta^{-1/2}J\Omega_{{\rm S}{\rm R},\beta,0}.$

The first equality in (3.27) is the defining property of $J$ and $\Delta$ and
the modular operator is $\Delta=e^{-\beta L_{0}}$. The second equality in
(3.27) follows from $JL_{0}=-L_{0}J$ [6, 12, 5, 29]. Next,

$\displaystyle\Delta^{1/2}\pi(X^{*})\Delta^{-1/2}$ $\displaystyle=$
$\displaystyle e^{{\rm i}tL_{0}}\pi(X^{*})e^{-{\rm i}tL_{0}}\big{|}_{t={\rm
i}\beta/2}$ (3.28) $\displaystyle=$ $\displaystyle\pi(e^{{\rm
i}tH_{0}}X^{*}e^{-{\rm i}tH_{0}})\big{|}_{t={\rm i}\beta/2}$ $\displaystyle=$
$\displaystyle\pi(e^{-\beta H_{0}/2}X^{*}e^{\beta H_{0}/2}).$

Using that $J\pi(\cdot)J=\pi^{\prime}(\cdot)$ and combining (3.28) with (3.27)
yields (3.26). Due to (3.26) we may replace, in (3.25), $\pi(K)$ by
$\pi^{\prime}(e^{-\beta H_{0}/2}K^{*}e^{\beta H_{0}/2})$ and use that the
latter operator commutes with $e^{{\rm i}tL_{\lambda}}\pi({O})e^{-{\rm
i}tL_{\lambda}}$,

$\displaystyle\rho_{{\rm S},\beta}\otimes\omega_{{\rm
R},\beta}\big{(}K^{*}\,{\rm e}^{{\rm i}tH_{\lambda}}{O}\,{\rm e}^{-{\rm
i}tH_{\lambda}}K\big{)}$ (3.29) $\displaystyle=\big{\langle}\Omega_{{\rm
S}{\rm R},\beta,0},\pi(K^{*})\pi^{\prime}(e^{-\beta H_{0}/2}K^{*}e^{\beta
H_{0}/2})e^{{\rm i}tL_{\lambda}}\pi({O})e^{-{\rm i}tL_{\lambda}}\Omega_{{\rm
S}{\rm R},\beta,0}\big{\rangle}.$

By (3.22) we have $e^{-{\rm i}tL_{\lambda}}\Omega_{{\rm S}{\rm
R},\beta,\lambda}=\Omega_{{\rm S}{\rm R},\beta,\lambda}$ and so

$e^{{\rm i}tL_{\lambda}}\pi({O})e^{-{\rm i}tL_{\lambda}}\Omega_{{\rm S}{\rm
R},\beta,0}=e^{{\rm i}tL_{\lambda}}\pi(O)\Omega_{{\rm S}{\rm
R},\beta,\lambda}+e^{{\rm i}tL_{\lambda}}\pi({O})\big{(}\Omega_{{\rm S}{\rm
R},\beta,0}-\Omega_{{\rm S}{\rm R},\beta,\lambda}\big{)},$ (3.30)

where the uncoupled equilibrium state $\Omega_{{\rm S}{\rm R},\beta,0}$ is
given in (3.14). We combine (3.29), (3.30) and (3.23) into

$\rho_{{\rm S},\beta}\otimes\omega_{{\rm R},\beta}\big{(}K^{*}\,{\rm e}^{{\rm
i}tH_{\lambda}}O\,{\rm e}^{-{\rm i}tH_{\lambda}}K\big{)}=\langle\phi,e^{{\rm
i}tL_{\lambda}}\psi\rangle+R_{1}(t,\lambda),$ (3.31)

where

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle\pi(K)\pi^{\prime}(e^{\beta
H_{0}/2}Ke^{-\beta H_{0}/2})\Omega_{{\rm S}{\rm R},\beta,0}$
$\displaystyle\psi$ $\displaystyle=$ $\displaystyle\pi(O)\Omega_{{\rm S}{\rm
R},\beta,0}$ (3.32)

and

$|R_{1}(\lambda,t)|\leq C|\lambda|\,\|O\|\,\big{\|}\pi(K)\pi^{\prime}(e^{\beta
H_{0}/2}Ke^{-\beta H_{0}/2})\Omega_{{\rm S}{\rm R},\beta,0}\big{\|},$ (3.33)

uniformly in $t\in\mathbb{R}$. Now we apply the results of [30] to the main
term on the right side of (3.31). That work gives an expansion of
$\langle\phi,e^{{\rm i}tL_{\lambda}}\psi\rangle$ for vectors $\phi$, $\psi$
satisfying

$\phi,\psi\in{\mathcal{D}},\quad\bar{L}_{\lambda}\phi,\bar{L}_{\lambda}\psi\in{\mathcal{D}},$
(3.34)

where $\bar{L}_{\lambda}=P_{\rm R}^{\perp}L_{\lambda}P_{\rm R}^{\perp}|_{{\rm
Ran}P_{\rm R}^{\perp}}$ and where $\mathcal{D}\subset\mathcal{H}_{\rm GNS}$ is
a suitable sense set which we present in (3.89) below. The orthogonal
projection $P_{\rm R}$ is defined as

$P_{\rm R}=\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm S}\otimes|\Omega_{\rm
R}\rangle\langle\Omega_{\rm R}|,\qquad P_{\rm R}^{\perp}=\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}-P_{\rm R}.$ (3.35)

We show that (3.34) holds in Section 3.5 and continue here the analysis using
the result of Theorem 2.1 of [30],

$\langle\phi,e^{{\rm i}tL_{\lambda}}\psi\rangle=\langle\phi,(e^{{\rm
i}t(M(\lambda)}\otimes P_{\rm R})\psi\rangle+\langle\phi,P^{\perp}_{\rm
R}e^{{\rm i}tP^{\perp}_{\rm R}L_{\lambda}P^{\perp}_{\rm R}}P^{\perp}_{\rm
R}\psi\rangle+R(\lambda,t),$ (3.36)

with a remainder satisfying

$\big{|}R(\lambda,t)\big{|}\leq C|\lambda|^{1/4}\,{\mathcal{A}}(\phi,\psi),$
(3.37)

for constants $C$ and $\mathcal{A}$ independent of $\lambda$ and $t\geq 0$. We
now explain the operator $M(\lambda)$ in (3.36). Denote by ${\mathcal{E}}_{0}$
the set of all differences of eigenvalues of $H_{\rm S}$ (i.e., the spectrum
of $L_{\rm S}$, (3.19)). Denote the eigenprojection of $L_{0}$ associated to
$e\in{\mathcal{E}}_{0}$ by $P_{e}=P_{{\rm S},e}\otimes P_{\rm R}$, where
$P_{{\rm S},e}$ is the eigenprojection of $L_{\rm S}$ and define the level
shift operator

$\Lambda_{e}=-P_{e}IP^{\perp}_{e}(L_{0}-e+{\rm i}0_{+})^{-1}IP_{e},$ (3.38)

where ${\rm i}0_{+}$ is the limit of ${\rm i}\epsilon$ as $\epsilon\rightarrow
0_{+}$ (the limit in (3.38) exists in operator norm, see [30, 31]). The
operator $\Lambda_{e}$ acts on ${\rm ran}P_{e}$. It is explained in detail in
[30] how $\Lambda_{e}$ determines the eigenvalues of $L_{\lambda}$ close to
$e$. Under the Fermi Golden Rule and simplicity conditions (A2a), (A2b), the
operator $M(\lambda)$ in (3.36) is given by

$M(\lambda)=L_{\rm
S}+\lambda^{2}\Lambda,\qquad\Lambda=\bigoplus_{e\in{\mathcal{E}}_{0}}\Lambda_{e}.$
(3.39)

The operators $L_{\rm S}$ and $\Lambda$ commute. Now that we have explained
the objects defining the dynamics ‘along $P_{\rm R}$’ in (3.36) we can analyze
it further. Writing

$\pi({O})=\sum_{j}\pi({O}_{\rm S}^{j}\otimes{O}_{\rm R}^{j})=\sum_{j}\pi_{\rm
S}({O}_{\rm S}^{j})\otimes\pi_{\rm R}({O}_{\rm R}^{j})$ (3.40)

and taking into account (3.32) and (3.35), it follows that

$\displaystyle\langle\phi,(e^{{\rm i}tM(\lambda)}\otimes P_{\rm
R})\psi\rangle$ $\displaystyle=$ $\displaystyle\sum_{j}\omega_{{\rm
R},\beta}({O}_{\rm R}^{j})\big{\langle}\phi,\big{(}e^{{\rm
i}tM(\lambda)}\pi_{\rm S}({O}_{\rm S}^{j})\Omega_{{\rm
S},\beta}\big{)}\otimes\Omega_{\rm R}\big{\rangle}$ (3.41) $\displaystyle=$
$\displaystyle\sum_{j}\omega_{{\rm R},\beta}({O}_{\rm
R}^{j})\big{\langle}\phi,\big{(}\pi_{\rm S}(e^{t{\mathcal{L}}_{*}}{O}_{\rm
S}^{j})\Omega_{{\rm S},\beta}\big{)}\otimes\Omega_{\rm R}\big{\rangle},$

where the operator ${\mathcal{L}}_{*}$ is uniquely defined by (see also [29,
30, 31])

$e^{{\rm i}tM(\lambda)}\pi_{\rm S}(X)\Omega_{{\rm S},\beta}=\pi_{\rm
S}\big{(}e^{t{\mathcal{L}}_{*}}X\big{)}\Omega_{{\rm S},\beta},\qquad\mbox{for
all system operators $X\in\mathcal{B}({\mathbb{C}^{N}})$}.$ (3.42)

The second factor of the summand of (3.41) is (c.f. (3.32))

$\displaystyle\big{\langle}\phi,\big{(}\pi_{\rm
S}(e^{t{\mathcal{L}}_{*}}{O}_{\rm S}^{j})\Omega_{{\rm
S},\beta}\big{)}\otimes\Omega_{\rm R}\big{\rangle}$ (3.43) $\displaystyle=$
$\displaystyle\big{\langle}\pi(K)\Omega_{{\rm S}{\rm
R},\beta,0},\pi\big{(}e^{t{\mathcal{L}}_{*}}{O}_{\rm
S}^{j}\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm R}\big{)}\pi^{\prime}(e^{-\beta
H_{0}/2}K^{*}e^{\beta H_{0}/2})\Omega_{{\rm S}{\rm R},\beta,0}\big{\rangle}$
$\displaystyle=$ $\displaystyle\rho_{{\rm S},\beta}\otimes\omega_{{\rm
R},\beta}\big{(}K^{*}(e^{t{\mathcal{L}}_{*}}{O}_{\rm
S}^{j}\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm R})K\big{)},$

where we used (3.24), (3.26) in the last step. Now we remember that we wrote
$K$ for $K_{\alpha}$, and summing over $\alpha$, as per (3.24), we obtain from
(3.31), (3.36),

$\displaystyle\rho_{{\rm S}{\rm R}}(e^{{\rm i}tH_{\lambda}}{O}e^{-{\rm
i}tH_{\lambda}})$ $\displaystyle=$
$\displaystyle\sum_{\alpha}\sum_{j}\omega_{{\rm R},\beta}({O}_{\rm
R}^{j})\,\rho_{{\rm S},\beta}\otimes\omega_{{\rm
R},\beta}\big{(}K_{\alpha}^{*}(e^{t{\mathcal{L}}_{*}}{O}_{\rm
S}^{j}\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm R})K_{\alpha}\big{)}$ (3.44)
$\displaystyle+\big{\langle}\Phi,P^{\perp}_{\rm R}e^{{\rm i}tP^{\perp}_{\rm
R}L_{\lambda}P^{\perp}_{\rm R}}P^{\perp}_{\rm R}\pi({O})\Omega_{{\rm S}{\rm
R},\beta,0}\big{\rangle}+R_{2}(\lambda,t),$

where

$\Phi=\sum_{\alpha=1}^{\nu}\pi(K_{\alpha})\pi^{\prime}(e^{\beta
H_{0}/2}K_{\alpha}e^{-\beta H_{0}/2})\Omega_{{\rm S}{\rm R},\beta,0}$ (3.45)

and the remainder is the sum of (3.33) and (3.37),

$\displaystyle|R_{2}(\lambda,t)|$ $\displaystyle\leq$ $\displaystyle
C|\lambda|^{1/4}\sum_{\alpha=1}^{\nu}{\mathcal{A}}\Big{(}\pi(K_{\alpha})\pi^{\prime}(e^{\beta
H_{0}/2}K_{\alpha}e^{-\beta H_{0}/2})\Omega_{{\rm S}{\rm
R},\beta,0}\,,\,\pi({O})\Omega_{{\rm S}{\rm R},\beta,0}\Big{)}$ (3.46)
$\displaystyle+C|\lambda|\,\|{O}\|\,\sum_{\alpha=1}^{\nu}\big{\|}\pi(K_{\alpha})\pi^{\prime}(e^{\beta
H_{0}/2}K_{\alpha}e^{-\beta H_{0}/2})\Omega_{{\rm S}{\rm R},\beta,0}\big{\|}.$

In terms of the reduced initial state $\rho_{\rm S}$, defined in (2.26), the
first term on the right side of (3.44) is simply

$\displaystyle\sum_{\alpha}\sum_{j}\omega_{{\rm R},\beta}({O}_{\rm R}^{j})\
\rho_{{\rm S},\beta}\otimes\omega_{{\rm
R},\beta}\big{(}K_{\alpha}^{*}(e^{t{\mathcal{L}}_{*}}{O}_{\rm
S}^{j}\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm R})K_{\alpha}\big{)}$ (3.47)
$\displaystyle=$ $\displaystyle\sum_{j}\rho_{\rm
S}\big{(}e^{t{\mathcal{L}}_{*}}{O}_{\rm S}^{j}\big{)}\omega_{{\rm
R},\beta}({O}_{\rm R}^{j})$ $\displaystyle=$
$\displaystyle\sum_{j}\big{(}(e^{t{\mathcal{L}}}\rho_{\rm
S})\otimes\omega_{{\rm R},\beta}\big{)}({O}_{\rm S}^{j}\otimes{O}_{\rm
R}^{j})=\big{(}(e^{t{\mathcal{L}}}\rho_{\rm S})\otimes\omega_{{\rm
R},\beta}\big{)}({O}),$

where ${\mathcal{L}}$ is the adjoint of ${\mathcal{L}}_{*}$ with respect to
the inner product $(X,Y)={\rm tr}_{\rm S}X^{*}Y$ of the space of system
operators. It is shown in the Appendix of [29] that $\mathcal{L}$ is the
Davies generator. Combine (3.47) and (3.44),

$\rho_{{\rm S}{\rm R}}(e^{{\rm i}tH_{\lambda}}Oe^{-{\rm
i}tH_{\lambda}})=\big{(}(e^{t{\mathcal{L}}}\rho_{\rm S})\otimes\omega_{{\rm
R},\beta}\big{)}(O)+\chi(\lambda,t,O)+R_{2}(\lambda,t),$ (3.48)

where (recall $\Phi$ is given in (3.45))

$\chi(\lambda,t,O)=\big{\langle}\Phi,P^{\perp}_{\rm R}e^{{\rm
i}tP^{\perp}_{\rm R}L_{\lambda}P^{\perp}_{\rm R}}P^{\perp}_{\rm
R}\pi(O)\Omega_{{\rm S}{\rm R},\beta,0}\big{\rangle}.$ (3.49)

This is the expansion (2.32) of Theorem 2.2. The decay (2.36) follows from the
regularity of the resolvent, as explained in (1.10) of [30] (see also the
related Lemma 3.1 below). Note that $P^{\perp}_{\rm R}\pi(O_{\rm
S}\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm R})\Omega_{{\rm S}{\rm
R},\beta,0}=0$. Using this in (3.49) shows (2.35). Furthermore, for initial
states $\rho_{{\rm R}{\rm S}}=\rho_{\rm S}\otimes\omega_{{\rm R},\beta}$ the
Kraus operators are of the form
$K_{\alpha}=K^{\prime}_{\alpha}\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm R}$ and thus
$P^{\perp}_{\rm R}\Phi=0$, see (3.45). Using this information in (3.49) shows
(2.34). We now prove (2.37). From (3.49),

$\chi(\lambda,t,O)=\langle\Phi,P_{\rm R}^{\perp}e^{{\rm
i}tL_{0}}\pi(O)\Omega_{{\rm S}{\rm R},\beta,0}\rangle+T(\lambda,t,O)$ (3.50)

where, writing $\bar{X}$ for $P_{\rm R}^{\perp}XP_{\rm R}^{\perp}|_{{\rm
Ran}P_{\rm R}^{\perp}}$,

$T(\lambda,t,O)={\rm i}\lambda\int_{0}^{t}\big{\langle}\Phi,P^{\perp}_{\rm
R}e^{{\rm i}s\bar{L}_{\lambda}}\bar{I}e^{-{\rm i}(s-t)L_{0}}P^{\perp}_{\rm
R}\pi(O)\Omega_{{\rm S}{\rm R},\beta,0}\big{\rangle}ds.$ (3.51)

Using that $P^{\perp}_{\rm R}=\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}-P_{\rm R}$ one readily
sees that

$\langle\Phi,P_{\rm R}^{\perp}e^{{\rm i}tL_{0}}\pi(O)\Omega_{{\rm S}{\rm
R},0}\rangle=\rho_{{\rm S}{\rm R}}\big{(}e^{{\rm i}tH_{0}}Oe^{-{\rm
i}tH_{0}}\big{)}-(\rho_{\rm S}\otimes\omega_{{\rm R},\beta})\big{(}e^{{\rm
i}tH_{0}}Oe^{-{\rm i}tH_{0}}\big{)}.$ (3.52)

In order to show the bound (2.37) it suffices to prove that for all $t\geq 0$,

$|T(\lambda,t,O)|\leq C|\lambda|.$ (3.53)

The clue in this bound is that the integrand of (3.51) decays in $s$
sufficiently quickly to be integrable. Let $A=d\Gamma({\rm i}\partial_{u})$,
acting on $\mathcal{H}_{\rm S}$ (3.3). Denote the $j$-fold commutator of
$\bar{X}$ with $\bar{A}$ by ${\rm
ad}^{j}_{\bar{A}}(\bar{X})=[\cdots[[\bar{X},\bar{A}],\bar{A}]\cdots]$.

###### Lemma 3.1

Suppose the form factor $g$ satisfies
$\|\partial_{u}^{j}\tau_{\beta}g\|_{L^{2}({\mathbb{R}}\times S^{2})}<\infty$
for $j=0,\ldots,4$ and suppose $X$ is an operator such that
$\|\,\mathcal{N}^{-1/2}{\rm
ad}^{j}_{\bar{A}}(\bar{X})\,\mathcal{N}^{-1/2}\|<\infty$ for $j=1,2,3$, where
$\,\mathcal{N}=d\Gamma(\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\mathcal{H}_{\rm R}})$ is the number
operator in the Fock space $\mathcal{H}_{\rm R}$ (see (3.3) and also (3.80)).
Then we have

$\max_{1\leq r\leq
3}\Big{|}\big{\langle}\Phi,(\bar{L}_{\lambda}-z)^{-1}\bar{X}(\bar{L}_{0}-\zeta)^{-r}\Psi\big{\rangle}\Big{|}\leq
C_{\bar{X}}\|\Phi\|_{3}\,\|\Psi\|_{3},$ (3.54)

uniformly in $z,\zeta$ with ${\rm Im}z$, ${\rm Im}\zeta<0$. Here,
$\|\cdot\|_{j}$ is the norm (3.88) and $C_{\bar{X}}$ is a constant depending
on $\bar{X}$.

We give a proof of Lemma 3.1 below. For now we apply it to estimate the term
(3.51). We write the propagator $e^{-{\rm i}(s-t)L_{0}}$ in its resolvent
representations (Fourier-Laplace transform, see also equation (1.30) in [30]),
for $w>0$,

$\displaystyle\big{\langle}\Phi,P^{\perp}_{\rm R}e^{{\rm
i}s\bar{L}_{\lambda}}\bar{I}e^{-{\rm i}(s-t)L_{0}}P^{\perp}_{\rm
R}\pi(O)\Omega_{{\rm S}{\rm R},\beta,0}\big{\rangle}$ (3.55) $\displaystyle=$
$\displaystyle\frac{-1}{2\pi{\rm i}}\int_{{\mathbb{R}}-{\rm i}w}\frac{e^{{\rm
i}(t-s)\zeta}}{(\zeta+{\rm i})^{2}}\langle\Phi,P^{\perp}_{\rm R}e^{{\rm
i}s\bar{L}_{\lambda}}\bar{I}(\bar{L}_{0}-\zeta)^{-1}P^{\perp}_{\rm
R}\Psi\rangle d\zeta,$

where $\Psi=(\bar{L}_{0}+{\rm i})^{2}\pi(O)\Omega_{{\rm S}{\rm R},\beta,0}$.
Next we proceed analogously for $e^{{\rm i}s\bar{L}_{\lambda}}$ to obtain (set
$\tau=t-s$)

$\displaystyle T(\lambda,t,O)$ $\displaystyle=$ $\displaystyle{\rm
i}\lambda\big{(}\frac{-1}{2\pi{\rm
i}}\big{)}^{2}\int_{0}^{t}d\tau\int_{\mathbb{R}-{\rm i}w}dz\frac{e^{{\rm
i}(t-\tau)z}}{(z+{\rm i})^{2}}\int_{{\mathbb{R}}-{\rm i}w}d\zeta\
\frac{e^{{\rm i}\tau\zeta}}{(\zeta+{\rm i})^{2}}$ (3.56)
$\displaystyle\times\langle\Phi^{\prime},(\bar{L}_{\lambda}-z)^{-1}\bar{I}(\bar{L}_{0}-\zeta)^{-1}\Psi\rangle,$

with $\Phi^{\prime}=(\bar{L}_{\lambda}+{\rm i})^{2}\Phi$. One shows that
$\Psi$ and $\Phi^{\prime}$ are well defined and
$\|\Phi^{\prime}\|_{3},\|\Psi\|_{3}<\infty$ by the arguments of Section 3.5.
We divide the $\tau$ integral into $\tau\in[0,1]$ and $\tau\in[1,t]$, so

$T(\lambda,t,O)=T_{1}(\lambda,t,O)+T_{2}(\lambda,t,O),$ (3.57)

where $T_{1}$ is given by (3.56) with $\int_{0}^{t}$ replaced by
$\int_{1}^{t}$ and $|T_{2}(\lambda,t,O)|\leq C|\lambda|$. To analyze $T_{1}$,
we use (twice) that $e^{{\rm i}\tau\zeta}=\frac{1}{{\rm
i}\tau}\partial_{\zeta}e^{{\rm i}\tau\zeta}$ and integrate by parts in
$\zeta$, to arrive at

$\displaystyle T_{1}(\lambda,t,O)$ $\displaystyle=$ $\displaystyle{\rm
i}\lambda\big{(}\frac{-1}{2\pi{\rm
i}}\big{)}^{2}\int_{1}^{t}d\tau\frac{-1}{\tau^{2}}\int_{\mathbb{R}-{\rm
i}w}dz\frac{e^{{\rm i}(t-\tau)z}}{(z+{\rm i})^{2}}\int_{{\mathbb{R}}-{\rm
i}w}d\zeta\ e^{{\rm i}\tau\zeta}$ (3.58)
$\displaystyle\times\partial^{2}_{\zeta}\big{\\{}(\zeta+{\rm
i})^{-2}\langle\Phi^{\prime},(\bar{L}_{\lambda}-z)^{-1}\bar{I}(\bar{L}_{0}-\zeta)^{-1}\Psi\rangle\big{\\}}.$

We now use Lemma 3.1 to bound the term $\partial^{2}_{\zeta}\\{(\zeta+{\rm
i})^{-2}\langle\Phi^{\prime},(\bar{L}_{\lambda}-z)^{-1}\bar{I}(\bar{L}_{0}-\zeta)^{-1}\Psi\rangle\\}$,
and we arrive at

$|T_{1}(\lambda,t,O)|\leq C|\lambda|\int_{1}^{t}\frac{d\tau}{\tau^{2}}\leq
C|\lambda|,\qquad\forall t\geq 1.$ (3.59)

Note that in the application of Lemma 3.1, we took $\bar{X}=\bar{I}$, and
$\|{\mathcal{N}}^{-1/2}{\rm
ad}_{\bar{A}}^{j}(\bar{I})\mathcal{N}^{-1/2}\|<\infty$ is guaranteed by the
condition 2.29 on the form factor $g$. This shows the relation (3.53) and
hence (2.37). The proof of Theorem 2.2 is complete, modulo a proof of Lemma
(3.1), which we now present.

Proof of Lemma 3.1.  As in the proof of Theorem 3.1 in [31], one considers the
regularization $\bar{L}_{\lambda}(\alpha)=\bar{L}_{0}+{\rm
i}\alpha\bar{\,}\mathcal{N}+\lambda\bar{I}(\alpha)$, $\alpha>0$, where

$\bar{I}(\alpha)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\widehat{f}(s)e^{{\rm
i}\alpha s\bar{A}}\bar{I}e^{-{\rm i}\alpha s\bar{A}}ds,\qquad A=d\Gamma({\rm
i}\partial_{u})$ (3.60)

and $\widehat{f}$ is the Fourier transform of a Schwartz function $f$
satisfying $f^{(k)}(0)=1$, $k=0,1,2,\ldots$ The bound (3.54) follows if the
same ($\alpha$ independent) bound can be shown with
$\bar{L}_{\lambda},\bar{L}_{0}$ replaced by
$\bar{L}_{\lambda}(\alpha),\bar{L}_{0}(\alpha)$, see Section 3.1.1 of [31].
Let us write for short $R_{z}=(\bar{L}_{\lambda}(\alpha)-z)^{-1}$ and
$S_{\zeta}=(\bar{L}_{0}(\alpha)-\zeta)^{-1}$. We also write $X$ instead of
$\bar{X}$ in the remainder of the proof, and $A$ for $\bar{A}$, $I$ for
$\bar{I}$. Note that $S_{\zeta}=e^{\alpha A}(L_{0}-\zeta)^{-1}e^{-\alpha A}$.
Using that $\partial_{\alpha}(S_{\zeta})^{k}=AS_{\zeta}^{k}-S_{\zeta}^{k}A$
and $\partial_{\alpha}R_{z}=AR_{z}-R_{z}A+\lambda R_{z}YR_{z}$, where

$Y=\partial_{\alpha}I(\alpha)-[A,I(\alpha)]=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}({\rm
i}s-1)\widehat{f}(s)e^{{\rm i}\alpha sA}[A,I]e^{-{\rm i}\alpha sA}ds,$ (3.61)

we get

$\displaystyle\partial_{\alpha}\langle\Phi,R_{z}XS_{\zeta}^{3}\Psi\rangle$
$\displaystyle=$
$\displaystyle\langle\Phi,AR_{z}XS_{\zeta}^{3}\Psi\rangle-\big{\langle}\Phi,R_{z}XS_{\zeta}^{3}A\Psi\rangle$
(3.62)
$\displaystyle+\langle\Phi,R_{z}[X,A]S_{\zeta}^{3}\Psi\big{\rangle}+\lambda\langle\Phi,R_{z}YR_{z}XS_{\zeta}^{3}\Psi\rangle.$

Using that $\|\,\mathcal{N}^{1/2}R_{z}A\Phi\|\leq C\alpha^{-1/2}\|\Phi\|_{2}$
(see Proposition 3.4(2) of [31]) and $\|R_{z}\|$, $\|S_{\zeta}\|\leq C/\alpha$
(see Proposition 3.3(1) of [31]), together with the fact that
$\|\,\mathcal{N}^{-1/2}Y\,\mathcal{N}^{-1/2}\|\leq C\alpha^{\ell}$, for any
$\ell=1,2,\ldots$ provided $\|\partial_{u}^{j}\tau_{\beta}g\|_{L^{2}}<\infty$
($j=0,\ldots,\ell+1$, see Proposition 3.4(1) of [31]333This convenient bound
is not hard to derive by expanding $e^{{\rm i}\alpha sA}[A,I]e^{-{\rm i}\alpha
sA}$ in (3.61) in a power series in $\alpha$ and using that $f^{(k)}(0)=1$.),
we estimate $|\langle\Phi,AR_{z}XS_{\zeta}^{3}\Psi\rangle|\leq
C\alpha^{-3}\|\,\mathcal{N}^{-1/2}X\,\mathcal{N}^{-1/2}\|\,\|\Phi\|_{2}\|\Psi\|_{2}$
and the same upper bound for the second term on the right side of (3.62). The
third one is estimated by
$|\langle\Phi,R_{z}[X,A]S_{\zeta}^{3}\Psi\rangle|\leq
C\alpha^{-3}\|\,\mathcal{N}^{-1/2}[X,A]\,\mathcal{N}^{-1/2}\|\,\|\Phi\|_{2}\|\Psi\|_{2}$.
For the fourth term on the right side, we take $\ell=1$ to get
$|\langle\Phi,R_{z}YR_{z}XS_{\zeta}^{3}\Psi\rangle|\leq
C\alpha^{-3}\|\,\mathcal{N}^{-1/2}X\,\mathcal{N}^{-1/2}\|\Phi\|_{1}\|\Psi\|_{1}$
(we also use the bound $\|\,\mathcal{N}^{1/2}R_{z}\,\mathcal{N}^{1/2}\|\leq
C$, c.f. Proposition 3.3(2) of [31]). All together, we obtain

$\big{|}\partial_{\alpha}\langle\Phi,R_{z}XS_{\zeta}^{3}\Psi\rangle\big{|}\leq
C\alpha^{-3}\|\Phi\|_{2}\|\Psi\|_{2}.$ (3.63)

We integrate in $\alpha$, using that
$|\langle\Phi,R_{z}XS^{3}_{\zeta}\Psi\rangle|_{\alpha=1}\leq
C\|\Phi\|\,\|\Psi\|$, to obtain

$\big{|}\langle\Phi,R_{z}XS_{\zeta}^{3}\Psi\rangle\big{|}\leq
C(1+\alpha^{-2})\|\Phi\|_{2}\|\Psi\|_{2}.$ (3.64)

Now we use the bound (3.64) to estimate the three first terms on the right
side of (3.62) from above by $C(1+\alpha^{-2})\|\Phi\|_{3}\|\Psi\|_{3}$,
provided that $\|\,\mathcal{N}^{-1/2}[[X,A],A]\,\mathcal{N}^{-1/2}\|<\infty$.
The norm is now $\|\cdot\|_{3}$ due to the presence of the operator $A$ in the
first two terms of the right side of (3.62). We use $\ell=2$ to estimate the
last term on the right side of (3.62) and hence we get (3.63) with
$\alpha^{-3}$ replaced by $\alpha^{-2}$. Thus (3.64) holds with $\alpha^{-2}$
replaced by $\alpha^{-1}$. We now repeat the process one more time, using the
latest bounds in (3.62) and integrating to obtain
$|\langle\Phi,R_{z}XS_{\zeta}^{3}\Psi\rangle|\leq C\|\Phi\|_{3}\|\Psi\|_{3}$.
It is assumed that $\ell=3$ and
$\|\,\mathcal{N}^{-1/2}[[[X,A],A],A]\,\mathcal{N}^{-1/2}\|<\infty$.

This concludes the proof of Lemma 3.1 and hence that of Theorem 2.2. $\Box$

###  3.4 Proof of Proposition 2.1

We first show Lemma 3.2 and 3.3 detailing some representation independent
bounds on polynomials of creation, annihilation and Weyl operators. Then give
the proof of Proposition 2.1, starting at (3.79) below.

The number operator on the Fock space ${\mathcal{F}}$ (2.4) is given by

$\widehat{N}=d\Gamma(\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm
1\mskip-5.0mul}_{\mathcal{F}})=\int_{{\mathbb{R}}^{3}}a^{*}(k)a(k)d^{3}k.$
(3.65)

We often write $\widehat{N}$ for $\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}_{{\mathbb{C}}^{N}}\otimes\widehat{N}$. In order to alleviate
the notation, in this section we will use the symbol $N$ for the number
operator $\widehat{N}$, even though $N$ also denotes the dimension of the
system Hilbert space.

###### Lemma 3.2

For any $K\in{\mathcal{X}}\cup{\mathcal{P}}$ and any $\epsilon>0$, we have
$\|Ke^{-\epsilon N}\|<\infty$.

Proof of Lemma 3.2.  Let $K=a^{(\sigma_{1})}(f_{1})\cdots
a^{(\sigma_{\ell})}(f_{\ell})\in{\mathcal{P}}$,where $\sigma_{j}\in\\{\pm
1\\}$ and $a^{(+1)}(f)=a^{*}(f)$, $a^{(-1)}(f)=a(f)$. Let $\xi(N)$ be a
function of $N$. Since $Na^{*}(f)=a^{*}(f)(N+1)$ we have
$\xi(N)a^{*}(f)=a^{*}(f)\xi(N+1)$ and similarly $\xi(N)a(f)=a(f)\xi(N-1)$, so
in short, $\xi(N)a^{(\sigma)}(f)=a^{(\sigma)}(f)\xi(N+\sigma)$. Taking
$\xi(N)=(N+\ell)^{1/2}$,

$\displaystyle K$ $\displaystyle=$ $\displaystyle
a^{(\sigma_{1})}(f_{1})(N+\ell)^{-1/2}\cdot(N+\ell)^{1/2}a^{(\sigma_{2})}(f_{2})\cdots
a^{(\sigma_{\ell})}(f_{\ell})$ (3.66) $\displaystyle=$ $\displaystyle
a^{(\sigma_{1})}(f_{1})(N+\ell)^{-1/2}\cdot a^{(\sigma_{2})}(f_{2})\cdots
a^{(\sigma_{\ell})}(f_{\ell})\big{(}N+\ell+\textstyle\sum_{j=2}^{\ell}\sigma_{j}\big{)}^{1/2}.$

Using $(N+\ell)^{-1/2}(N+\ell)^{1/2}=\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}$ next to every
$a^{(\sigma_{j})}(f_{j})$ and pulling $(N+\ell)^{1/2}$ to the right as in
(3.66) gives

$K=a^{(\sigma_{1})}(f_{1})(N+\ell)^{-1/2}\cdots
a^{(\sigma_{\ell})}(f_{\ell})(N+\ell)^{-1/2}\cdot B,$ (3.67)

where

$\displaystyle B$ $\displaystyle=$
$\displaystyle(N+\ell)^{1/2}(N+\ell+\sigma_{\ell-1})^{1/2}(N+\ell+\sigma_{\ell-1}+\sigma_{\ell-2})^{1/2}\cdots(N+\ell+\textstyle\sum_{j=2}^{\ell}\sigma_{j})^{1/2}$
(3.68) $\displaystyle\leq$ $\displaystyle(N+2\ell)^{\ell/2}.$

We know that $\|a^{(\sigma)}(f)(N+1)^{-1/2}\|\leq\|f\|_{L^{2}}$, see e.g. [6,
32]. It follows from (3.67) and (3.68),

$\|Ke^{-\epsilon N}\|\leq\|f_{1}\|_{L^{2}}\cdots\|f_{\ell}\|_{L^{2}}\
\sup_{n\in{\mathbb{N}}}(n+2\ell)^{\ell/2}e^{-\epsilon n}.$ (3.69)

Now for $0<\epsilon<1/4$ (which we can take without loss of generality), we
have

$\sup_{n\in{\mathbb{N}}}(n+2\ell)^{\ell/2}e^{-\epsilon
n}\leq\Big{(}\frac{\ell}{2\epsilon}\Big{)}^{\ell/2}$ (3.70)

and so it follows from (3.69) and (3.70) that the statement of Lemma 3.2 holds
for $K$ a product of creation and annihilation operators and hence for all
$K\in{\mathcal{P}}$. Next let

$K=e^{{\rm i}\sum_{r=1}^{R}B_{r}\otimes a^{\sharp}(f_{r})}\in{\mathcal{X}}.$
(3.71)

We expand the exponential,

$Ke^{-\epsilon N}=\sum_{\ell=0}^{\infty}\frac{{\rm
i}^{\ell}}{\ell!}\big{[}\sum_{r=1}^{R}B_{r}\otimes
a^{\sharp}(f_{r})\big{]}^{\ell}e^{-\epsilon N}.$ (3.72)

We then multiply out the product $[\sum_{r=1}^{R}B_{r}\otimes
a^{\sharp}(f_{r})]^{\ell}$ into a sum of $R^{\ell}$ terms, each term being an
element of the form $H\otimes Q\in{\mathcal{P}}$, where
$H\in{\mathcal{B}}({\mathbb{C}}^{N})$ and $Q$ is a product of $\ell$ creation
and annihilation operators. By (3.69), (3.70), $\|Qe^{-\epsilon N}\|\leq
C^{\ell}\,(\ell/(2\epsilon))^{\ell/2}$ for some constant $C$. The norm of the
general term in the series (3.72) is bounded above by

$\Big{\|}\frac{{\rm i}^{\ell}}{\ell!}\big{[}\sum_{r=1}^{R}B_{r}\otimes
a^{\sharp}(f_{r})\big{]}^{\ell}e^{-\epsilon N}\Big{\|}\leq
C^{\ell}\frac{1}{\ell!}\Big{(}\frac{\ell}{2\epsilon}\Big{)}^{\ell/2}$ (3.73)

for another constant $C$ (depending on $R$) and the series converges for any
$\epsilon>0$ (and any $R$), as is easily established using e.g. the ratio
test. This shows that (3.72) is a bounded operator. This concludes the proof
of Lemma 3.2. $\Box$

The domain of definition of the operator $e^{\epsilon N}$, denoted ${\rm
Dom}(e^{\epsilon N})$, is a dense set in ${\mathcal{F}}$.

###### Lemma 3.3

Given any $\epsilon>0$, all operators in $\mathfrak{C}$ are well defined on
${\rm Dom}(e^{\epsilon N})$.

Proof of Lemma 3.3.  To control products of operators, we note that $\forall
s\in{\mathbb{R}}$,

$\displaystyle e^{sN}a^{\sharp}(f)e^{-sN}$ $\displaystyle=$ $\displaystyle
e^{\pm s}a^{\sharp}(f)$ (3.74) $\displaystyle e^{sN}e^{{\rm
i}\sum_{r=1}^{R}B_{r}\otimes a^{\sharp}(f_{r})}e^{-sN}$ $\displaystyle=$
$\displaystyle e^{{\rm i}\sum_{r=1}^{R}e^{\pm s}\,B_{r}\otimes
a^{\sharp}(f_{r})},$ (3.75)

where the $+s$ exponent is present for $a^{\sharp}=a^{*}$ and the $-s$
exponent for $a^{\sharp}=a$. Equations (3.74), (3.75) show that

$e^{sN}{\mathfrak{C}}e^{-sN}={\mathfrak{C}},\qquad\forall s\in{\mathbb{R}}.$
(3.76)

Let now $K_{\alpha}\in{\mathcal{X}}\cup{\mathcal{P}}$, $\alpha=1,\ldots\nu$.
Then $\forall\epsilon>0$,

$\displaystyle K_{1}\cdots K_{\nu}$ (3.77) $\displaystyle=$ $\displaystyle
K_{1}e^{-\epsilon N}\,(e^{\epsilon N}K_{2}e^{-2\epsilon N})\,(e^{2\epsilon
N}K_{3}e^{-3\epsilon N})\cdots(e^{(\nu-1)\epsilon N}K_{\nu}e^{-\nu\epsilon
N})e^{\nu\epsilon N}.$

For $\ell\geq 1$ integer, $e^{(\ell-1)\epsilon N}K_{\ell}e^{-\ell\epsilon
N}=e^{(\ell-1)\epsilon N}K_{\ell}e^{-(\ell-1)\epsilon N}e^{-\epsilon N}\equiv
K_{\ell}(\epsilon)e^{-\epsilon N}$, where
$K(\epsilon)\in{\mathcal{X}}\cup{\mathcal{P}}$ is the $K$ with all single-
particle functions $f,g$ replaced by $e^{(\ell-1)\epsilon}f$ and
$e^{-(\ell-1)\epsilon}g$ for creation and annihilation operators,
respectively, according to (3.74), (3.75). As shown in Lemma 3.2, we have
$\|K_{\ell}(\epsilon)e^{-\epsilon N}\|<\infty$ for any $\epsilon>0$. It
follows that

$K_{1}\cdots K_{\nu}=K_{1}e^{-\epsilon N}K_{2}(\epsilon)e^{-\epsilon
N}K_{3}(\epsilon)e^{-\epsilon N}\cdots K_{\nu}(\epsilon)e^{-\epsilon
N}\,e^{\nu\epsilon N}$ (3.78)

is a well defined operator on ${\rm Dom}(e^{\nu\epsilon N})$, for any
$\epsilon>0$. This completes the proof of Lemma 3.3. $\Box$

Proof of Proposition 2.1.  To prove Proposition 2.1, we consider

$\rho_{{\rm S},\beta}\otimes\omega_{{\rm R},\beta}(K^{*}e^{{\rm
i}tH_{\lambda}}Oe^{-{\rm i}tH_{\lambda}}K)=\langle\Omega_{{\rm S}{\rm
R},\beta,0},\pi(K^{*})e^{{\rm i}tL_{\lambda}}\pi(O)e^{-{\rm
i}tL_{\lambda}}\pi(K)\Omega_{{\rm S}{\rm R},\beta,0}\rangle$ (3.79)

for some $K\in{\mathfrak{C}}$ and $O\in{\mathcal{O}}$. We only need to show
the following result.

###### Lemma 3.4

We have $\|\pi(K)\Omega_{{\rm S}{\rm R},\beta,0}\|<\infty$ for any
$K\in\mathfrak{C}$.

Proof of Lemma 3.4. According to (3.12) the operator $\pi(a^{*}(f))$ is the
sum of a creation plus an annihilation operator on the Fock space
${\mathcal{H}}_{\rm R}$, (3.3), and so we have
$\|\pi_{\beta}(a^{*}(f))(\,\mathcal{N}+1)^{-1/2}\|<\infty$. Here,
$\,\mathcal{N}$ is the number operator on the Fock space ${\mathcal{H}}_{\rm
R}$, (3.3),

$\,\mathcal{N}=d\Gamma(\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{{\mathcal{H}}_{\rm R}}).$ (3.80)

We can then proceed entirely analogously to the proof of Lemma 3.2 to show
that for all $K\in{\mathcal{P}}$, we have
$\|\pi(K)e^{-\epsilon\,\mathcal{N}}\|<\infty$, for any $\epsilon>0$. Next, let
$K$ be given by (3.71). From (3.5), (3.8), (3.12) we get

$\pi\big{(}B_{r}\otimes a^{\sharp}(f_{r})\big{)}=B_{r}\otimes\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}_{{\rm
S}}\otimes\big{(}a_{\beta}^{*}(f_{r,+})+a_{\beta}(f_{r,-})\big{)},$ (3.81)

for functions $f_{r,+}$, $f_{r,-}\in L^{2}({\mathbb{R}}\times S^{2})$. It
follows that

$\pi\big{(}e^{{\rm i}\sum_{r=1}^{R}B_{r}\otimes
a^{\sharp}(f_{r})}\big{)}=e^{{\rm i}\sum_{r=1}^{R}B_{r}\otimes\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}_{\rm S}\otimes[a_{\beta}^{*}(f_{r,+})+a_{\beta}(f_{r,-})]}.$
(3.82)

Expanding the exponential as in (3.72) and proceeding exactly as in the proof
of Lemma 3.2 we see that $\|\pi(e^{{\rm i}\sum_{r=1}^{R}B_{r}\otimes
a^{\sharp}(f_{r})})e^{-\epsilon\,\mathcal{N}}\|<\infty$ for all $\epsilon>0$.

So far in this proof, we have shown that

$\|\pi(K)e^{-\epsilon\,\mathcal{N}}\|<\infty,\qquad\forall\epsilon>0,\quad\
\forall K\in{\mathcal{X}}\cup{\mathcal{P}}.$ (3.83)

To see that $\|\pi(K_{1}\cdots K_{\nu})\Omega_{{\rm S}{\rm
R},\beta,0}\|<\infty$ we proceed as in (3.77), inserting factors
$\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}=e^{-\ell\epsilon\,\mathcal{N}}e^{\ell\epsilon\,\mathcal{N}}$,

$\pi(K_{1}\cdots
K_{\nu})=\pi(K_{1})e^{-\epsilon\,\mathcal{N}}\,(e^{\epsilon\,\mathcal{N}}\pi(K_{2})e^{-2\epsilon\,\mathcal{N}})\cdots(e^{(\nu-1)\epsilon\,\mathcal{N}}\pi(K_{\nu})e^{-\nu\epsilon\,\mathcal{N}})e^{\nu\epsilon\,\mathcal{N}}.$
(3.84)

All that is left to do is showing that for any $\ell\geq 1$ integer,

$\|e^{(\ell-1)\epsilon\,\mathcal{N}}\pi(K)e^{-\ell\epsilon\,\mathcal{N}}\|<\infty,\qquad\forall\epsilon>0,\
\forall K\in{\mathcal{X}}\cup{\mathcal{P}}.$ (3.85)

We do this as above in Lemma 3.3. Indeed, we have for all $s\in\mathbb{R}$,

$\displaystyle e^{s\,\mathcal{N}}a_{\beta}^{\sharp}(v)e^{-s\,\mathcal{N}}$
$\displaystyle=$ $\displaystyle e^{\pm s}a_{\beta}(v),\qquad v\in
L^{2}({\mathbb{R}}\times S^{2})$ $\displaystyle
e^{s\,\mathcal{N}}\pi\big{(}e^{{\rm i}\sum_{r=1}^{R}B_{r}\otimes
a^{\sharp}(f_{r})}\big{)}e^{-s\,\mathcal{N}}$ $\displaystyle=$ $\displaystyle
e^{{\rm i}\sum_{r=1}^{R}B_{r}\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm
S}\otimes[e^{s}a_{\beta}^{*}(f_{r,+})+e^{-s}a_{\beta}(f_{r,-})]},$ (3.86)

where again, $+s$ and $-s$ are for the creation and the annihilation operator,
respectively, as in (3.74), (3.75). The bound (3.85) now follows from (3.83)
just as in the proof of Lemma 3.3. This shows that

$\|\pi(K)e^{-\epsilon\,\mathcal{N}}\|<\infty,\quad\|e^{(\ell-1)\epsilon\,\mathcal{N}}\pi(K)e^{-\ell\epsilon\,\mathcal{N}}\|<\infty,\qquad\forall\ell\in{\mathbb{N}},\epsilon>0,K\in{\mathfrak{C}}.$
(3.87)

This concludes the proof of Lemma 3.4 and hence the proof of Proposition (2.1)
is complete. $\Box$

###  3.5 Set of regular vectors $\mathcal{D}$ and proof of (3.34)

As in [31] we define the norms $\|\cdot\|_{j}$, for $j=0,1,2,\ldots$, on
$\mathcal{H}_{\rm GNS}$ (3.2):

$\|\phi\|_{j}=\|(1+\bar{A}^{2})^{j/2}\phi\|,\qquad A=d\Gamma({\rm
i}\partial_{u}),\qquad\bar{A}=P^{\perp}_{\rm R}AP^{\perp}_{\rm R}|_{{\rm
Ran}P_{\rm R}^{\perp}}.$ (3.88)

We have $\|\phi\|_{k}\leq\|\phi\|_{\ell}$ for $0\leq k\leq\ell$. The dense set
${\mathcal{D}}\subset{\mathcal{H}}_{\rm GNS}$ is given by

${\mathcal{D}}=\\{\phi\in{\mathcal{H}}_{\rm GNS}\ :\ \|\phi\|_{3}<\infty\\}.$
(3.89)

We now verify that (3.34) holds, where the two vectors $\phi$, $\psi$ are
given by

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle\pi(K)\pi^{\prime}(e^{\beta
H_{0}/2}Ke^{-\beta H_{0}/2})\Omega_{{\rm S}{\rm R},\beta,0},$
$\displaystyle\psi$ $\displaystyle=$ $\displaystyle\pi(O)\Omega_{{\rm S}{\rm
R},\beta,0}.$ (3.90)

We first show that $\|\phi\|_{3}<\infty$. We have

$\displaystyle e^{\beta H_{0}/2}(B\otimes a^{\sharp}(f))e^{-\beta H_{0}/2}$
$\displaystyle=$ $\displaystyle e^{\beta H_{\rm S}/2}Be^{-\beta H_{\rm
S}/2}\otimes a^{\sharp}(e^{\pm\beta|k|/2}f)$ (3.91) $\displaystyle e^{\beta
H_{0}/2}e^{{\rm i}\sum_{r=1}^{R}B_{r}\otimes a^{\sharp}(f_{r})}e^{-\beta
H_{0}/2}$ $\displaystyle=$ $\displaystyle e^{{\rm i}\sum_{r=1}^{R}e^{\beta
H_{\rm S}/2}B_{r}e^{-\beta H_{\rm S}/2}\otimes
a^{\sharp}(e^{\pm\beta|k|/2}f_{r})}$ (3.92)

where plus sign in the exponentials in (3.91), (3.92) is present if
$a^{\sharp}=a^{*}$ and the minus sign if $a^{\sharp}=a$. Note that
$e^{\pm\beta H_{\rm S}/2}$ is bounded and $\,{\rm e}^{\pm\beta|k|/2}f_{r}\in
L^{2}$, so that the operators (3.91) and (3.92) have the same structure as the
operators in ${\mathfrak{C}}$ (execpt that here, the single-particle functions
$f$ appear with the weights $e^{\pm\beta|k|/2}$). It follows that analyzing
$\phi$ in (3.90) reduces to analyzing vectors of the form
$\phi^{\prime}=\pi(K)\pi^{\prime}(K^{\prime})\Omega_{{\rm S}{\rm R},\beta,0}$,
where $K,K^{\prime}\in{\mathfrak{C}}_{0}$, and where $\mathfrak{C}_{0}$ is the
algebra defined by (c.f. (2.22))

${\mathfrak{C}}_{0}={\rm LinSpan}\big{\\{}K_{1}\cdots K_{n}\ :\
K_{j}\in{\mathcal{X}}_{0}\cup{\mathcal{P}}_{0}\ :\ n\in{\mathbb{N}}\big{\\}},$
(3.93)

where ${\mathcal{X}}_{0}$ and ${\mathcal{P}}_{0}$ are as $\mathcal{X}$ and
$\mathcal{P}$ given in (2.21) and (2.20), but with $L^{2}_{\rm cor}$ replaced
by $L^{2}_{\rm obs}$ (we do not assume $e^{\beta|k|}f\in
L^{2}({\mathbb{R}}^{3},d^{3}k)$). Using that $J\,\mathcal{N}=\,\mathcal{N}J$
and $\,\mathcal{N}\Omega_{{\rm S}{\rm R},\beta,0}=0$, we get

$\displaystyle\pi(K)\pi^{\prime}(K^{\prime})\Omega_{{\rm S}{\rm R},\beta,0}$
$\displaystyle=$ $\displaystyle\pi(K)J\pi(K^{\prime})J\Omega_{{\rm S}{\rm
R},\beta,0}$ (3.94) $\displaystyle=$
$\displaystyle\pi(K)e^{-\epsilon\,\mathcal{N}}Je^{\epsilon\,\mathcal{N}}\pi(K^{\prime})e^{-2\epsilon\,\mathcal{N}}J\Omega_{{\rm
S}{\rm R},\beta,0}.$

The relation (3.94) together with (3.87) (which holds for
$K\in{\mathfrak{C}}_{0}$) then shows immediately that
$\|\pi(K)\pi^{\prime}(K^{\prime})\Omega_{{\rm S}{\rm R},\beta,0}\|<\infty$,
for all $K,K^{\prime}\in{\mathfrak{C}}_{0}$. In other words,
$\|\phi\|_{0}<\infty$, where this is the norm (3.88) with $j=0$ of the vector
$\phi$, (3.90). We now show that $\|\phi\|_{3}<\infty$. Writing simply $A$ for
$\bar{A}$ (see (3.88)) in the following argument, we have

$\|\phi\|^{2}_{j}=\langle\phi,(1+A^{2})^{j}\phi\rangle\leq
C\langle\phi,(1+A^{2j})\phi\rangle=C\big{(}\|\phi\|^{2}+\|A^{j}\phi\|^{2}\big{)},$
(3.95)

for a constant $C$ independent of $\phi$. It follows that $\|\phi\|_{j}\leq
C(\|\phi\|+\|A^{j}\phi\|)$. We thus only need to show that
$\|A^{3}\phi\|<\infty$, that is, we need to show that

$\|A^{3}\pi(K)\pi^{\prime}(K^{\prime})\Omega_{{\rm S}{\rm
R},\beta,0}\|<\infty,\quad\mbox{ for all $K,K^{\prime}\in{\mathfrak{C}}_{0}$.
}$ (3.96)

We have for all $\alpha\in\mathbb{R}$ (see also (3.82)),

$e^{{\rm i}\alpha A}\pi(e^{{\rm i}\sum_{r=1}^{R}B_{r}\otimes
a^{\sharp}(f_{r})})e^{-{\rm i}\alpha A}=e^{{\rm
i}\sum_{r=1}^{R}B_{r}\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm
S}\otimes[a_{\beta}^{*}(e^{-\alpha\partial_{u}}f_{r,+})+a_{\beta}(e^{-\alpha\partial_{u}}f_{r,-})]}.$
(3.97)

Applying the operation $-{\rm i}\partial_{\alpha}|_{\alpha=0}$ to the left
side of (3.97) gives the commutator of $A$ with $\pi(e^{{\rm
i}\sum_{r=1}^{R}B_{r}\otimes a^{\sharp}(f_{r})})$. To calculate the
corresponding right hand side, we recall that well-known formula for an
operator family $Y(\alpha)$ depeding on $\alpha\in\mathbb{R}$,

$\partial_{\alpha}e^{Y(\alpha)}=\int_{0}^{1}e^{sY(\alpha)}\big{(}\partial_{\alpha}Y(\alpha)\big{)}e^{(1-s)Y(\alpha)}ds.$
(3.98)

Thus $-{\rm i}\partial_{\alpha}|_{\alpha=0}$ applied to both sides of (3.97)
results in the expression

$A\pi(e^{{\rm i}\sum_{r=1}^{R}B_{r}\otimes a^{\sharp}(f_{r})})-\pi(e^{{\rm
i}\sum_{r=1}^{R}B_{r}\otimes a^{\sharp}(f_{r})})A={\mathcal{C}},$ (3.99)

where the commutator is

$\displaystyle{\mathcal{C}}=-\int_{0}^{1}\pi(e^{{\rm
i}s\sum_{r=1}^{R}B_{r}\otimes a^{\sharp}(f_{r})})$ (3.100)
$\displaystyle\quad\times\Big{(}\sum_{r=1}^{R}B_{r}\otimes\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}_{\rm
S}\otimes[a^{*}_{\beta}(\partial_{u}f_{r,+})+a_{\beta}(\partial_{u}f_{r,-})]\Big{)}\pi(e^{{\rm
i}(1-s)\sum_{r=1}^{R}B_{r}\otimes a^{\sharp}(f_{r})})ds.\ \ $

Of course, $\mathcal{C}$ is not a bounded operator, as usual, but by (3.87),
we have

$\|{\mathcal{C}}e^{-\epsilon\,\mathcal{N}}\|<\infty,\quad\|e^{\epsilon\,\mathcal{N}}{\mathcal{C}}e^{-2\epsilon\,\mathcal{N}}\|<\infty,\qquad\forall\epsilon>0.$
(3.101)

Next, using that $Aa^{*}_{\beta}(v)-a^{*}_{\beta}(v)A=a^{*}_{\beta}({\rm
i}\partial_{u}v)$ for any $v\in L^{2}({\mathbb{R}}\times S^{2})$, we see that
for any polynomial $P\in{\mathcal{P}}_{0}$, we have
$A\pi(P)-\pi(P)A={\mathcal{C}}$ for an operator $\mathcal{C}$ satisfying again
(3.101). Let now $L_{1},\ldots,L_{n}$ each be an operator belonging to
${\mathcal{X}}_{0}\cup{\mathcal{P}}_{0}$. Then $A\pi(L_{1}\cdots
L_{n})=\pi(L_{1})A\pi(L_{2}\cdots L_{n})+{\mathcal{C}_{1}}\pi(L_{2}\cdots
L_{n})$, and keeping up commuting $A$ to the right through the operators $L$
gives

$A\pi(L_{1}\cdots L_{n})=\pi(L_{1}\cdots L_{n})A+{\mathcal{C}},$ (3.102)

where ${\mathcal{C}}$ satisfies (3.101). Now since
$\pi^{\prime}(\cdot)=J\pi(\cdot)J$ and $JA=AJ$, we get with the same argument
that

$A\pi^{\sharp}(L_{1})\cdots\pi^{\sharp}(L_{n})=\pi^{\sharp}(L_{1})\cdots\pi^{\sharp}(L_{n})A+{\mathcal{C}},$
(3.103)

with ${\mathcal{C}}$ satisfying (3.101). Here, each $\pi^{\sharp}$
individually is either $\pi$ or $\pi^{\prime}$. Then by linearity, the formula
(3.103) is also correct if each $L_{j}\in{\mathfrak{C}}_{0}$ (as opposed to
being just an element of ${\mathcal{X}}_{0}\cup{\mathcal{P}}_{0}$).

We now combine (3.103) with the fact that $A\Omega_{{\rm S}{\rm R},\beta,0}=0$
to conclude that (3.96) holds true for $A^{3}$ replaced by $A$. In other
words, we have $\|\phi\|_{1}<\infty$. But applying $A$ to both sides of
(3.103) gives

$\displaystyle A^{2}\pi^{\sharp}(L_{1})\cdots\pi^{\sharp}(L_{n})$
$\displaystyle=$ $\displaystyle
A\pi^{\sharp}(L_{1})\cdots\pi^{\sharp}(L_{n})A+A{\mathcal{C}}_{1}$ (3.104)
$\displaystyle=$
$\displaystyle\pi^{\sharp}(L_{1})\cdots\pi^{\sharp}(L_{n})A^{2}+{\mathcal{C}}_{2}A+A{\mathcal{C}}_{1}.$

We can commute $A$ through ${\mathcal{C}}_{1}$,
$A{\mathcal{C}}_{1}={\mathcal{C}}_{1}A+{\mathcal{C}}^{\prime}$. The operator
${\mathcal{C}}^{\prime}$ once again will satisfy
$\|{\mathcal{C}}^{\prime}e^{-\epsilon\,\mathcal{N}}\|<\infty$, any
$\epsilon>0$. This is easily seen since ${\mathcal{C}}_{1}$ is a sum of
products of elements in ${\mathcal{P}}_{0}$ and operators of the form (3.100).
One may then proceed as above to show the bound
$\|{\mathcal{C}}^{\prime}e^{-\epsilon\,\mathcal{N}}\|<\infty$, any
$\epsilon>0$. This shows that $\|\phi\|_{2}<\infty$. We repeat the procedure
and apply $A$ to (3.104) to conclude that $\|\phi\|_{3}<\infty$. The only
limitation on the number of times we can repeat the procedure is that
successive derivatives $\partial_{u}$ of all functions $f_{r,+}$, $f_{r,-}$
involved should stay in $L^{2}({\mathbb{R}}\times S^{2})$. Each additional
application of $A$ requires one more such derivative to be $L^{2}$. For
$\|\phi\|_{j}<\infty$ we need the derivatives up to and including order $j$ to
be square integrable. This condition for $j=0,\ldots,3$ is guaranteed by
taking functions $f(k)$ from the set $L^{2}_{\rm cor}$, see after (2.18).

This shows that $\|\phi\|_{3}<\infty$. Next we need to show that
$\|\bar{L}_{\lambda}\phi\|_{3}<\infty$ (see (3.34)). We have

$A^{3}L_{\lambda}=L_{0}A^{3}+3{\rm
i}\,\mathcal{N}A^{2}+\lambda\big{(}IA^{3}+3I_{1}A^{2}+3I_{2}A+I_{3}\big{)},$
(3.105)

where $I_{k}$ is the $k$ fold commutator of $I$ with $A$. We have
$\|I(\,\mathcal{N}+1)^{-1/2}\|<\infty$ and
$\|I_{k}(\,\mathcal{N}+1)^{-1/2}\|<\infty$. It follows that
$\|\bar{L}_{\lambda}\phi\|_{3}<\infty$ provided that

$\|\,\mathcal{N}A^{j}\phi\|<\infty,\ j=0,\ldots,3\quad\mbox{and}\quad\|L_{\rm
R}A^{3}\phi\|<\infty.$ (3.106)

By the above arguments, we know that
$\|e^{\epsilon\,\mathcal{N}}\phi\|<\infty$ and
$\|e^{\epsilon\,\mathcal{N}}A^{3}\phi\|<\infty$ for all $\epsilon>0$, so the
bounds involving $\,\mathcal{N}$ in (3.106) hold. To show that
$A^{3}\phi\in{\rm Dom}(L_{\rm R})$ we repeat the argument after (3.99) above.
Instead of commuting $A$ through elements of ${\mathfrak{C}}_{0}$, now we have
commute $L_{\rm R}$ through them. We have $L_{\rm
R}a^{\sharp}_{\beta}(v)-a^{\sharp}_{\beta}(v)L_{\rm R}=\pm
a_{\beta}^{\sharp}(uv)$, where the plus sign is for
$a_{\beta}^{\sharp}=a_{\beta}^{*}$ and $uv$ is the function $u\cdot
v(u,\Sigma)$. The multiplication by $u$ of the functions $v(u,\Sigma)$
preserves the square integrability as guaranteed by the the definition of
$L^{2}_{\rm obs}$, $L^{2}_{\rm cor}$ (2.18). The same argument as above then
shows that $\|L_{\rm R}A^{3}\phi\|<\infty$.

We have shown so far that $\phi,\bar{L}_{\lambda}\phi\in\mathcal{D}$. To
finish the proof of (3.34) we also need to show that
$\psi,\bar{L}_{\lambda}\psi\in\mathcal{D}$, where $\psi=\pi(O)\Omega_{{\rm
S}{\rm R},\beta,0}$, see (3.32). According to (3.9) we have

$\pi_{\beta}(W(f))=e^{{\rm i}\varphi_{\beta}(\tau_{\beta}f)}=e^{{\rm
i}[a_{\beta}^{*}(\tau_{\beta}f)+a_{\beta}(\tau_{\beta}f)]/\sqrt{2}}\in{\mathfrak{C}}_{0}.$
(3.107)

It follows that ${\mathcal{O}}\subset{\mathfrak{C}}_{0}$ and hence showing
$\psi,\bar{L}_{\lambda}\psi\in\mathcal{D}$ is a special case of the proof that
$\phi,\bar{L}_{\lambda}\phi\in\mathcal{D}$. This completes the proof of
(3.34). $\Box$

###  3.6 Level shift operators

The level shift operators $\Lambda_{e}$ are defined in (3.38). As $I$ contains
two terms, see (3.21), $\Lambda_{e}$ is the sum of four terms,

$\displaystyle\Lambda_{e}=$ $\displaystyle-P_{e}\\{G\otimes\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}_{\rm S}\otimes\varphi_{\beta}(\tau_{\beta}g)\\}(L_{0}-e+{\rm
i}0_{+})^{-1}\\{G\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm
S}\otimes\varphi_{\beta}(\tau_{\beta}g)\\}P_{e}$
$\displaystyle+P_{e}\\{G\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm
S}\otimes\varphi_{\beta}(\tau_{\beta}g)\\}(L_{0}-e+{\rm
i}0_{+})^{-1}\\{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm
S}\otimes{\mathcal{C}}G{\mathcal{C}}\otimes\varphi_{\beta}(e^{-\beta
u/2}\tau_{\beta}g)\\}P_{e}$ $\displaystyle+P_{e}\\{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}_{\rm
S}\otimes{\mathcal{C}}G{\mathcal{C}}\otimes\varphi_{\beta}(e^{-\beta
u/2}\tau_{\beta}g)\\}(L_{0}-e+{\rm i}0_{+})^{-1}\\{G\otimes\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}_{\rm S}\otimes\varphi_{\beta}(\tau_{\beta}g)\\}P_{e}$
$\displaystyle-P_{e}\\{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm
S}\otimes{\mathcal{C}}G{\mathcal{C}}\otimes\varphi_{\beta}(e^{-\beta
u/2}\tau_{\beta}g)\\}(L_{0}-e+{\rm i}0_{+})^{-1}\\{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}_{\rm
S}\otimes{\mathcal{C}}G{\mathcal{C}}\otimes\varphi_{\beta}(e^{-\beta
u/2}\tau_{\beta}g)\\}P_{e}.$

The partial trace over the reservoir part is calculated using the formula

$\displaystyle P_{\rm R}\varphi_{\beta}(F)(L_{0}-e+{\rm
i}0_{+})^{-1}\varphi_{\beta}(G)P_{\rm R}=\tfrac{1}{2}P_{\rm
R}a_{\beta}(F)(L_{0}-e+{\rm i}0_{+})^{-1}a^{*}_{\beta}(G)P_{\rm R}$ (3.109)
$\displaystyle\qquad\qquad\qquad=\tfrac{1}{2}P_{\rm R}\int_{{\mathbb{R}}\times
S^{2}}\bar{F}(u,\Sigma)G(u,\Sigma)(L_{\rm S}-e+u+{\rm i}0_{+})^{-1}dud\Sigma,$

valid for any two functions $F,G\in L^{2}({\mathbb{R}}\times S^{2})$. One
obtains from (3.6)

$\displaystyle\Lambda_{e}=$ $\displaystyle-\tfrac{1}{2}P_{{\rm
S},e}(G\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm S})\int_{{\mathbb{R}}\times
S^{2}}\frac{\big{|}g(|u|,\Sigma)\big{|}^{2}}{|1-e^{-\beta u}|}(L_{\rm
S}-e+u+i0_{+})^{-1}u^{2}dud\Sigma\ (G\otimes\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}_{\rm S})P_{{\rm S},e}$ $\displaystyle+\tfrac{1}{2}P_{{\rm
S},e}(G\otimes\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm S})\int_{{\mathbb{R}}\times
S^{2}}\frac{e^{-\beta u/2}\big{|}g(|u|,\Sigma)\big{|}^{2}}{|1-e^{-\beta
u}|}(L_{\rm S}-e+u+i0_{+})^{-1}u^{2}dud\Sigma\ (\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}_{\rm S}\otimes{\mathcal{C}}G{\mathcal{C}})P_{{\rm S},e}$
$\displaystyle+\tfrac{1}{2}P_{{\rm S},e}(\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm
S}\otimes{\mathcal{C}}G{\mathcal{C}})\int_{{\mathbb{R}}\times
S^{2}}\frac{e^{-\beta u/2}\big{|}g(|u|,\Sigma)\big{|}^{2}}{|1-e^{-\beta
u}|}(L_{\rm S}-e+u+i0_{+})^{-1}u^{2}dud\Sigma\ (G\otimes\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}_{\rm S})P_{{\rm S},e}$ $\displaystyle-\tfrac{1}{2}P_{{\rm
S},e}(\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm
S}\otimes{\mathcal{C}}G{\mathcal{C}})\int_{{\mathbb{R}}\times
S^{2}}\frac{\big{|}g(|u|,\Sigma)\big{|}^{2}}{|e^{\beta u}-1|}(L_{\rm
S}-e+u+i0_{+})^{-1}u^{2}dud\Sigma\ (\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}_{\rm
S}\otimes{\mathcal{C}}G{\mathcal{C}})P_{{\rm S},e}.$

The condition that $\Lambda_{e}$ should have simple spectrum can then be
verified for concrete cases, where $g,G$ are given explicitly. Note that if
$e$ is a simple eigenvalue of $L_{\rm S}$, then $P_{{\rm S},e}$ has rank one
and so $\Lambda_{e}$ has automatically simple spectrum. Note also that $e=0$
is always a degenerate eigenvalue of $L_{\rm S}$. The explicit form of the
level shift operators for several models can be found in [26, 31]. Additional
information on their structure is provided in [28].

Acknowledgements.  The author was supported by a Discovery Grant from NSERC,
the National Sciences and Engineering Research Council of Canada.

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