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counting statistics, Fredholm determinants
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\title{Fredholm determinants and the statistics of charge transport}

\begin{document}
\author{ J.~E.~Avron ${}^{(a)}$, S. Bachmann ${}^{(b)}$, G.M. Graf ${}^{(b)}$
and I. Klich ${}^{(c)}$\\
\normalsize\it
${}^{(a)}$ Department of Physics, Technion, 32000 Haifa, Israel\\
\normalsize\it
${}^{(b)}$ Theoretische Physik, ETH-H\"onggerberg, 8093 Z\"urich, Switzerland\\
\normalsize\it
${}^{(c)}$ Condensed Matter Department,
Caltech, MC 114-36, Pasadena, CA 91125, USA}
\maketitle
%%%%%%%%%%%
\begin{abstract} Using operator algebraic methods we
show that the moment generating function of charge transport in a
system with infinitely many non-interacting Fermions is given by a
determinant of a certain operator in the one-particle Hilbert
space. The formula is equivalent to a formula of Levitov in the
finite dimensional case and may be viewed as its regularized form
in general. Our result embodies two fundamental principles in
mesoscopic physics, namely, that the transport properties are
essentially independent of the length of the leads and of the
depth of the Fermi sea.
\end{abstract}

\section{Introduction}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Models of physical systems are often formulated with the help of one
or few parameters which guarantee that whatever one computes is well
defined and finite while, at the same time, are believed not to
affect properties of physical interest. Examples are: The number of
particles in a macroscopic system, and the lattice spacing
(ultraviolet cutoffs) in the study of critical phenomena.

The theory of transport in mesoscopic systems has two such
parameters: The length of the incoming leads that connect to the
system and the depth of the Fermi sea. The independence of the
length of the leads is the statement that well designed
experiments measure the transport properties of the mesoscopic
system and are independent of the measuring circuit. The
independence of the depth of the Fermi sea expresses the
irrelevance for transport of electrons that are buried deep in the
Fermi sea, 
since in most situations they can not be excited above it. In this
sense there is freedom from both the volume and the the ultraviolet scale. 

One strategy to address this type of behavior is to consider
idealized systems where the parameters are taken to be infinitely
large. The limiting idealized system comes with the
price tag that expressions for physical quantities that are
otherwise guaranteed to be finite, may become ambiguous, formal
and even infinite. The value in worrying about this idealized,
possibly un-physical system, is precisely in that once the
ambiguities and infinities are resolved, they teach us something
important about the finite physical model, namely, that the
parameters used in its formulation, do indeed effectively
disappear from the physical properties. Their role is effectively
reduced to the control the small differences between the idealized
model and the physical one.


We shall consider a problem of this kind that arises in the
context of modeling the statistics of charge transport from one
reservoir to another. Levitov and Lesovik \cite{Levitov}
wrote a formula for the appropriate generating function in terms
of a certain infinite dimensional determinant. When one wants to
apply this formula to the idealized cases one finds ambiguities
and, as emphasized by Levitov et al. 
\cite{LevitovLeeLesovik, Ivanov, Levitov2}, the determinant
requires proper definition through regularization. We intend to further the
understanding of these points by providing an alternative,
mathematically consistent, form for the determinant. As we
shall see, the ``regularized form'' of the determinant naturally
emerges once the quantum dynamics is formulated on the state space
of the idealized system. 

In the next section we introduce the statistics of charge transport, review
the Levitov determinant, and propose a regularization. In 
Section~\ref{Results} we state the main results. Section~\ref{Proofs} is 
devoted to proofs and Section~\ref{exa} exemplifies the assumptions made 
in this work.

%%%%%%%%%%%%

\section{Levitov's formula and its regularization}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We consider a lead, where independent electrons are evolved over
some time interval and ask about the statistics of the charge
transferred from the left to the right portion of the lead. 
To begin, we recall the result obtained
in \cite{Levitov} and further elaborated in
\cite{LevitovLeeLesovik,Ivanov}. 
We present its derivation and
generalization to finite times along the heuristic lines given in
\cite{Klich}, in the sense that we do as if the one-particle
Hilbert space $\hilb$ were finite-dimensional.

The fermionic Fock space $\mathcal{F}$ over $\hilb$ contains a
distinguished state, the vacuum, with the physical interpretation
of a no particle state. Let $\Tr$, resp. $\tr$, denote the trace
on $\mathcal{F}$, resp. $\hilb$. Let $U$ be the unitary on $\hilb$
representing the time evolution, and $Q$ the projection
corresponding to the right portion of the lead. Their second
quantizations, $\Gamma(U)=\wedge_{i=1}^k U_i$, resp.
$\dd\Gamma(Q)=\sum_{i=1}^k Q_i$ on $k$-particle states, then stand
for the evolution on $\mathcal{F}$, resp. for the charge in that
portion. We suppose that the initial many particle (mixed) state
is of the form
\begin{equation*}
P=Z^{-1}\Gamma(M)
\end{equation*}
for some operator $M\geq0$, where $Z=\Tr\,\Gamma(M)=\det(1+M)$
ensures that $\Tr P=1$. The reduced one-particle
density matrix $N$ is defined by the property that
\begin{equation*}
\tr(AN)=\Tr\left(\dd\Gamma(A)P\right)
\end{equation*}
for any one-particle operator $A$ on $\hilb$. In our case,
$N=M(\Id+M)^{-1}$. This follows from
\begin{align}
\Tr \left(\ep{\iu\lambda\dd\Gamma(A)}P\right)&=\Tr \left(\Gamma(\ep{\iu\lambda A})P\right)=Z^{-1}\Tr\left(\Gamma(\ep{\iu\lambda A}M)\right)=\frac{\det(\Id+\ep{\iu\lambda A}M)}{\det (\Id+M)} \nonumber \\ &=\det(\Id-N+\ep{\iu\lambda A}N)
\label{TrDet}
\end{align}
by taking the derivative at $\lambda=0$. 

In the following, we assume that $M$ and $Q$, and hence
$P$ and $\dd\Gamma(Q)$, commute, which physically means that in
the state defined by $P$, charge in the lead measured by $Q$ is a
good quantum number. Hence
\begin{equation*}
P|\alpha\rangle=\rho_{\alpha}|\alpha\rangle\,,\qquad\dd\Gamma(Q)|\alpha\rangle=
n_{\alpha}|\alpha\rangle\,,
\end{equation*}
for some basis $\{|\alpha\rangle\}$ of $\mathcal{F}$. The
moment generating function for the charge transfer statistics is
\begin{equation*}
\chi(\lambda)=\sum_{n\in\mathbb{Z}}p_n \ep{\iu\lambda n}\,,
\end{equation*}
where $p_n$ is the probability for $n$ electrons being deposited
into the right portion of the lead by the end of the time
interval. It may be computed as a sum over initial and final
states, $\alpha$ resp. $\beta$, with the former weighted according
to their probabilities $\rho_{\alpha}$:
\begin{align}
\label{BaddefChi}
\chi(\lambda)&=\sum_{\alpha,\beta}|\langle\beta|\Gamma(U)|\alpha\rangle|^2\,\rho_{\alpha}\ep{\iu\lambda(n_{\beta}-n_{\alpha})}=\Tr\left(\Gamma(U)\str\ep{\iu\lambda\dd\Gamma(Q)}\Gamma(U)\ep{-\iu\lambda\dd\Gamma(Q)}P\right) \\
&=Z^{-1}\Tr\left(\Gamma(U\str\ep{\iu\lambda Q}U\ep{-\iu\lambda Q}M)\right)=
\det\left(\Id-N+\ep{\iu\lambda U\str Q U}N\ep{-\iu\lambda Q}\right)\,,
\nonumber
\end{align}
where the trace has been computed in the basis $|\alpha\rangle$, with an
identity $\sum|\beta\rangle\langle\beta|=\Id$ absorbed at the
left of $\Gamma(U)$; the last equality is by~(\ref{TrDet}). This
is Levitov's formula:
\begin{equation}
\label{Levitov}
\chi(\lambda)=\det D(\lambda)\,,\qquad D(\lambda)=N'+\ep{\iu\lambda Q_U}N\ep{-\iu\lambda Q}\,,
\end{equation}
with $N'=\Id-N$ and $Q_U=U\str Q U$. Since $Q$ is a projection,
$\ep{2\pi i Q}=\ep{2\pi i Q_U}=1$ and $D(\lambda )$ is a periodic
function with period $2\pi$. This expresses the integrality of
charge transport.

An example of a state of interest is that of a system at inverse temperature
$\beta$ having one-particle Hamiltonian $H$; it is $P=Z^{-1}\Gamma(M)$ with
$M=\exp(-\beta H)$ and $N=[\Id+\exp(\beta H)]^{-1}$. In the limit
$\beta\to\infty$, $P$ describes the Fermi sea, whence $N$ is the projection
onto the occupied one-particle states.

The above derivation would be rigorous if the one-particle Hilbert space 
were finite dimensional. The question we want to address here is what is the
correct replacement for $D(\lambda)$ when $P$ describes infinitely
many particles, both
because the lead may be infinitely extended spatially (as
appropriate for an open system) and because the Fermi sea may be
very or even infinitely deep. The first concern
appears to affect only the derivation, but not the result,
eq.~(\ref{Levitov}). However, by the second, $D(\lambda)$ differs
from the identity by more than a trace class operator, as would be
required by the definition of a Fredholm determinant. A
manifestation thereof (and in a sense the only one) is that the
expected charge transport
\begin{equation}
\label{ExpTransp}
\langle n \rangle=-\iu\chi'(0)=
-\iu\left.\frac{d}{d\lambda}\det D(\lambda)\right|_{\lambda=0}=
\tr\left((Q_U-Q)N\right)
\end{equation}
involves an operator which is not trace class in the stated situation.
These statements are
illustrated (in the $\beta=\infty$ case) in Fig.~\ref{fig1} representing the
phase space of a single particle moving freely.

\begin{figure}[h]
\begin{center}
%\input{figure1.pdf_t}
\input{figure1.pstex_t}
\caption{Left: dispersion relation $E(p)$ of free particles, and
its linearization. Right: phase space (coordinates $x$, $p$) with
regions selected by $N$, $Q$ and $Q_U$.}\label{fig1}
\end{center}
\end{figure}

The Fermi sea $N$ corresponds to $|p|<p_F$, $p_F$ being the Fermi
momentum, and similarly the right half of the lead $Q$ to $x>0$.
The free evolution, which we take as a simple example for $U$, is
a horizontal shear, so that $Q_U-Q$ is associated with two
sectors, labelled $+$ and $-$. Their intersection with the
horizontal strip associated with $N$ delineates the phase space
support of $(Q_U-Q)N$. Its area, which is a rough estimate of the
trace class norm of the operator, is proportional to the depth of
the sea. If the dispersion relation is conveniently linearized at
$\pm p_F$, the depth becomes infinite, implying that the operator
is not trace class. As a remedy, we note that the expression
\begin{equation*}
\tr\left(QN-Q_UN_U\right)=0
\end{equation*}
vanishes by splitting the trace, though only suggestively so, because the
traces fail to exist separately due to the infinite spatial extent of the
leads. Adding nevertheless that expression to~(\ref{ExpTransp}) yields
\begin{equation}\label{nav}
\langle n \rangle=\tr\left(Q_U(N-N_U)\right)\,,
\end{equation}
which vanishes in the special case of the free evolution, $N_U=N$, and is
expected to be finite in others. This way of
renormalizing the expression is actually declaring that the 
Fermi sea does not
contribute to the current, instead of relying on a compensation between left
and right movers, as indicated by $+$ and $-$ in the figure.

This heuristic manipulation motivates the following regularization of the
Levitov determinant. Replacing $D(\lambda)$ by
\begin{equation}
\label{tildeLevitov}
\widetilde{D}(\lambda)=\ep{-\iu\lambda N_U Q_U}D(\lambda)\ep{\iu\lambda N Q}
\end{equation}
should not change the value of the determinant, since informally
\begin{equation}
\det(\ep{-\iu\lambda N_U Q_U})\cdot\det(\ep{\iu\lambda N
  Q})=\ep{\iu\lambda\,\tr(QN-Q_UN_U)}=1\,.
\label{formal}
\end{equation}
Moreover, this regularization affects only the first cumulant of the
statistics, \ie to the average charge transfer, since the full set of
cumulants is generated
by $\log \widetilde{D}(\lambda)$. From eq.~(\ref{Levitov}) we thus find
\begin{gather}
\label{nicedet}
\chi(\lambda)=\det\widetilde{D}(\lambda)\,, \\
\label{defnicedet}
\widetilde{D}(\lambda)=\ep{-\iu\lambda N_U Q_U}N'\ep{\iu\lambda N Q}+\ep{\iu\lambda N'_U Q_U}N\ep{-\iu\lambda N' Q}\,.
\end{gather}
It is to be noted that this representation of $\chi(\lambda)$ is
manifestly particle-hole symmetric:
\begin{equation}
\label{PHsymm}
\chi_N(\lambda)=\chi_{N'}(-\lambda)\,.
\end{equation}
It is also $2\pi$-periodic in $\lambda$, though manifestly so only at
$T=0$ since $NQ$, $N'Q$ etc. are all projections.
%(The case of finite temperatures is treated in corollary 2.) 
In that case, eq.~(\ref{defnicedet}) reduces to 
\begin{equation*}
\widetilde{D}(\lambda)=
1+Q_U(N-N_U)\bigl((\ep{\iu\lambda}-1)N-(\ep{-\iu\lambda}-1)N'\bigr)\,,
\end{equation*}
which shows that the generating function $\chi(\lambda)$ is well-defined
whenever its first cumulant (\ref{nav}) is. As we shall see, a 
slightly weaker result holds at positive temperature. 

Let us mention a few connections to other work. A related regularization 
of Levitov's determinant at zero
temperature was used in \cite{MuzykantskyAdamov}, where the
relation of counting statistics to a Riemann-Hilbert problem was
studied. Another one, exhibiting the symmetry (\ref{PHsymm}),
was proposed in \cite{PilgramButtiker}. 
On the more mathematical
side, regularizations of determinants have been related to
renormalization in \cite{ESe}, though by means of a somewhat different 
regularization known as 
$\det_n(1+A)=\det(1+A)\exp{(\tr \sum_{j=1}^{n-1}(-1)^jA^j/j)}$. 
Extensive use of C*-algebras in the theory of open systems was recently made by
Jak{\v{s}}i{\'c} and Pillet, see \eg \cite{JP}.

The purpose of this work is to
show that, under reasonable assumptions, eq.~(\ref{nicedet})
is obtained {\em without recourse to regularizations}, if the
second quantization is built upon the Fermi sea rather than on the
vacuum $N=0$. 

%%%%%%%%%%

\section{Results}\label{Results}

Let $\hilb$ be a separable Hilbert space with the following
operators acting on it: An orthogonal projection $Q$, a unitary
$U$, and a selfadjoint $N$, with
\begin{equation}
\label{stateOp}
0\leq N\leq 1\,,
\end{equation}
whose physical interpretations have been described in the previous section. Let
$N'=\Id-N$. We denote by $\ideal{p}$, ($p\ge 1$) the Schatten trace ideals,
\ie the space of all bounded operators $A$ on $\hilb$ such that
$\|A\|_p^p:=\tr|A|^p<\infty$.

The algebra of canonical anticommutation relations (CAR) over $\hilb$ is the
C*-algebra $\alg{\hilb}$ generated by $\Id$, and the elements $a(f)$
and $a\str(f)$, ($f\in\hilb$), such that
\begin{enumerate}
\item the map $f\longmapsto a(f)$ is antilinear
\item $a\str(f)=a(f)\str$
\item these elements satisfy the following anticommutation relations
\begin{equation*}
\{a(f),a\str(g)\}=(f,g)\Id\,,
\end{equation*}
all other anticommutators vanishing.
\end{enumerate}

A (global) gauge transformation is expressed by the automorphism
$\alpha_\lambda: a(f)\mapsto a(\ep{\iu\lambda}f)$. A state
$\omega$ on $\alg{\hilb}$ is gauge-invariant if
$\omega(\alpha_\lambda(A))= \omega(A)$ for all $A\in \alg{\hilb}$.
The operator $N$ defines a gauge-invariant
quasi-free state $\omega_N$ through
\begin{equation}
\omega_N(a\str(f_n)\ldots a\str(f_1)\,a(g_1)\ldots
a(g_m))=\delta_{nm}\det(g_i, N f_j)\,,
\label{state}
\end{equation}
or equivalently by $\omega_N(a\str(f)a(g))=(g, N f)$ and Wick's lemma.
Let $(\hilb_{N}, \pi_N, \Omega_N)$ be the cyclic representation of $\omega_N$:
%
\begin{equation}
\label{GNS}
\omega_N(A)=(\Omega_N, \pi_N(A)\Omega_N)\,,\qquad (A\in \alg{\hilb})\,.
\end{equation}
%
The algebra of observables is the (strong) closure of the range of $\pi_N$,
which is equal to its double commutant
$\overline{\pi_N(\alg{\hilb})}=\pi_N(\alg{\hilb})''$. We also recall that a
state is pure if and only if $\pi_N(\alg{\hilb})$ is irreducible, \ie
$\pi_N(\alg{\hilb})'=\left\{c\cdot\Id\mid c\in\mathbb{C}\right\}$, see \eg
\cite{BratRob}, Thm.~2.3.19. This is equivalent to $N$ being a projection 
operator.

These concepts briefly reviewed, we are now ready to state our
main theorem. Its significance is discussed below in a series of
remarks. 
The key result, which is part (v) together with Corollary~\ref{cor},
states that the moment generating function is given by the regularized
determinant, as described in the previous section. 

\begin{thm}
\label{thm1}
Assume that
%
\begin{gather}
\label{QN}
[Q,N]=0\,, \\
\label{NTrCl}
\sqrt{N}-\sqrt{N_{U\str}}\,,\quad\sqrt{N'}-\sqrt{N'_{U\str}}\in\ideal{1}\,,
\end{gather}
where $N_{U\str}=UN{U\str}$.

\noindent {\rm (Pure state)} Suppose $N=N^2$. Then we have
\begin{enumerate}
\item $\widetilde{D}(\lambda)-\Id\in\ideal{1}$, where
$\widetilde{D}(\lambda)$ is given in eq.~(\ref{defnicedet}).
\item The Bogoliubov automorphisms induced on $\alg{\hilb}$ by the unitary 
operators $U$ and \linebreak $\exp(\iu\lambda Q)$
are implementable on $\hilb_N$: There exist a unitary operator
$\widehat{U}$ and a selfadjoint $\widehat{Q}$ on $\hilb_N$ such that
\begin{align}
\label{UImpl}
\widehat{U}\pi_N\bigl(a^{\#}(f)\bigr)\widehat{U}\str&= \pi_N\bigl(a^{\#}(Uf)\bigr)\,, \\
\label{QImpl}
\ep{\iu\lambda \widehat{Q}}\pi_N\bigl(a^{\#}(f)\bigr)\ep{-\iu\lambda
  \widehat{Q}}&= \pi_N\bigl(a^{\#}(\ep{\iu\lambda Q}f)\bigr)\,,
\end{align}
for all $f\in\hilb$.
\item $\ep{\iu\lambda \widehat{Q}}\in\pi_N\bigl(\alg{\hilb}\bigr)''$.
More
generally, $f(\widehat{Q})\in\pi_N\bigl(\alg{\hilb}\bigr)''$ for any bounded
function $f$.
\item The above properties define $\widehat{U}$ uniquely up to left multiplication with an element from
$\pi_N\bigl(\alg{\hilb}\bigr)'$, and $\widehat{Q}$ up to an additive
constant.
In particular, $\widehat{U}\str \ep{\iu\lambda
\widehat{Q}}\widehat{U} \ep{-\iu\lambda \widehat{Q}}$ is unaffected
by the ambiguities.
\item
\begin{equation}
\label{Main}
(\Omega_N,\widehat{U}\str\ep{\iu\lambda \widehat{Q}}\widehat{U} \ep{-\iu\lambda \widehat{Q}}\Omega_N)=\det \widetilde{D}(\lambda)\,.
\end{equation}
\end{enumerate}

\noindent {\rm (Mixed state)} The above conclusions hold
also for $0<N<\Id$ if, in addition,
%
\begin{equation}
\label{mixedcond}
Q\sqrt{NN'}\in\ideal{1}\,.
\end{equation}
%
\end{thm}

\noindent
{\bf Remarks.}
1)  Eq.~(\ref{NTrCl}) demands that the evolution $U$ preserves $N$,
    except for
   creating excitations within an essentially finite region in space and
   energy, as can be seen from the phase space picture given in the
   introduction. This assumption is appropriate for the evolution induced by a
   compact device operating smoothly during a finite time interval.

2)  The operators $\widehat{U}$, $\widehat{Q}$ in (ii) are
    replacements for the
   non-existent $\Gamma(U)$ and $\dd\Gamma(Q)$ mentioned in the
   introduction. Eqs.~(\ref{UImpl}, \ref{QImpl}) state that any additional
   particle in the system evolves by $U$, resp. contributes to the 
   charge as described by $Q$. 
%In particular, the l.h.s. of eq.~(\ref{Main}) is the adequate replacement
%   for~(\ref{BaddefChi}).

3) If the state is pure, the pair of equations~(\ref{NTrCl}) reduce
to the first one with square roots dropped, and property (iii) holds
trivially, since $\mathcal{B}(\hilb_N)=\pi_N(\alg{\hilb})''$,
the bounded operators on $\hilb_N$. Moreover, $\widehat{U}$ is
unique up to a phase. Incidentally, condition~(\ref{mixedcond})
would be trivial in this case.


4) Property (iii) states that $\widehat{Q}$ is an observable, and
the same is true for $\widehat{U}\str\widehat{Q}\widehat{U}$,
because of $\widehat{U}\str\pi_N(\alg{\hilb})\widehat{U}\subset
\pi_N(\alg{\hilb})$, see~(\ref{UImpl}). Thus, the total charges
before and after the evolution are separately bestowed with physical
meaning.

5) The physical origin of the extra assumption~(\ref{mixedcond})
needed in the mixed state case is as follows. In both cases, the
expected charge contained in a portion of the lead is of order of
its length $L$, or zero if renormalized by subtraction of a
background charge. In the pure case however, the Fermi sea is an
eigenvector of the charge operator, while for the mixed state, the
variance of the charge must itself be of order $L$, because the
occupation of the one-particle states is fluctuating, since
$NN'\neq0$. Hence, in this latter situation, the measurement of the
renormalized charge yields finite values only as long as $L$ is
finite, of which eq.~(\ref{mixedcond}) is a mathematical
abstraction. This condition, while unnecessary for property (ii), is
essential for (iii). Without the latter, the l.h.s. of
eq.~(\ref{Main}) appears to be ambiguous. On the other hand,
the weaker condition
%
\begin{equation}
(Q_U-Q)\sqrt{NN'}\in\ideal{1}\,,
\label{weaker}
\end{equation}
%
is sufficient for property (i) and to ensure that the difference
%$\widehat{U}\str\widehat{Q}\widehat{U}-\widehat{Q}$
$\widehat{Q_U-Q}$ is an observable.

6) The theorem does not apply to the general
case~(\ref{stateOp}). The two
  cases considered suffice for thermal states with $\beta=\infty$ and
  $0<\beta<\infty$.\\


Let $\widehat{Q}=\int n\, dP(n)$ be the spectral representation of
$\widehat{Q}$. According to quantum mechanical principles an ideal 
measurement of $\widehat{Q}$ with outcome $n$ in $dn$ collapses 
$\Omega_N$ to the state $dP(n)\Omega_N$, normalized to the probability 
$(dP(n)\Omega_N, dP(n)\Omega_N)$ of that outcome. Effectively, this means 
that $dP(n)\Omega_N$ is the state relevant for a second measurement. The
charge transfer is inferred from two measurements \cite{MuzykantskyAdamov} of
the charge $\widehat{Q}$, one before and one after the evolution of the 
system by 
$\widehat{U}$. The joint probability for measurements $n$ and $m$ is 
$(\widehat{U}dP(n)\Omega_N,dP(m) \widehat{U}dP(n)\Omega_N)$ and the
generating function appropriately defined as
\begin{equation*}
\chi_N(\lambda)=\iint(dP(n)\Omega_N, \widehat{U}^*dP(m)\widehat{U}dP(n)\Omega_N)
\ep{\iu\lambda(m-n)}\,.
\end{equation*}
%
\begin{cor}\label{cor}
The generating function is
%
\begin{equation*}
\chi_N(\lambda)=(\Omega_N, \widehat{U}\str\ep{\iu\lambda\widehat{Q}}\widehat{U}\ep{-\iu\lambda\widehat{Q}}\Omega_N)
\end{equation*}
%
and describes the transport of integer charges $n$ with non-negative
probabilities: 
%
\begin{equation*}
\chi_N(\lambda)=\sum_{n\in\mathbb{Z}}p_n\ep{\iu\lambda n}\,,\qquad
p_n\geq0 \,,\quad\sum_{n\in\mathbb{Z}}p_n=1\,.
\end{equation*}
%
Moreover, the particle-hole symmetry~(\ref{PHsymm}) holds true.
\end{cor}

%%%%%%%%%%

\section{Proofs}\label{Proofs}
We begin by giving the proof of the corollary and continue with that of part
(i) of the theorem. Then we give some preliminaries needed for the other
parts. Thereafter we prove parts (ii-iv) and, finally, (v).

\subsection{Proof of Corollary \ref{cor}}
We begin by recalling that every gauge-invariant state is a factor state (see
\cite{PS}, Thm.~5.1), \ie
%
\begin{equation}
\label{factor}
\pi_N(\alg{\hilb})'\cap\pi_N(\alg{\hilb})''=\{c\cdot 1 |\,c\in\mathbb{C}\}\,.
\end{equation}
%
{From} eq.~(\ref{QImpl}) and $\ep{2\pi\iu Q}=1$, we see that
$\ep{2\pi\iu \widehat{Q}}\in\pi_N(\alg{\hilb})'$, while by~(iii)
we have $\ep{2\pi\iu \widehat{Q}}\in\pi_N(\alg{\hilb})''$.
Thus $\ep{2\pi\iu \widehat{Q}}=c$, $(|c|=1)$ and we may assume
$c=1$ by adding an additive constant to $\widehat{Q}$, see~(iv).
The spectral representation of $\widehat{Q}$ is then of the form
%
\begin{equation}
\label{QSpectrDec}
\widehat{Q}=\sum_{n\in\mathbb{Z}} nP_n\,.
\end{equation}
%
We note that
%
\begin{equation}
\label{someExp}
(\Omega_N, \ep{\iu\lambda\widehat{Q}}A\ep{-\iu\lambda\widehat{Q}}\,\Omega_N)=
(\Omega_N, A\,\Omega_N)
\end{equation}
for $A\in\pi_N(\alg{\hilb})''$. Indeed, for $A=\pi_N(a\str(f)\,a(g))$, we have
$\ep{\iu\lambda\widehat{Q}}A\ep{-\iu\lambda\widehat{Q}}=
\pi_N(a\str(\ep{\iu\lambda Q}f)$ $a(\ep{\iu\lambda Q}g))$ by~(\ref{QImpl}).
The expectations~(\ref{someExp}) agree because of
$(\ep{\iu\lambda Q} g, N \ep{\iu\lambda Q} f)=(g, N f)$
by $[Q,N]=0$. The same holds true by~(\ref{state}) for arbitrary products of
$a\str(f_i)$, $a(g_i)$, and by density, for
$A\in\pi_N(\alg{\hilb})''$. By~(iii) we may apply~(\ref{someExp}) to
$A\ep{\iu\lambda\widehat{Q}}$ instead of $A$, and obtain $(\Omega_N,
P_n A\,\Omega_N)=(\Omega_N, AP_n\,\Omega_N)$; then this to $A P_n \in\pi_N(\alg{\hilb})''$ instead of $A$, and get $(\Omega_N,
A P_n\,\Omega_N)=(\Omega_N, P_n A P_n\,\Omega_N)$. Moreover, we have
$\widehat{U}\str\ep{\iu\lambda\widehat{Q}}\widehat{U}\in\widehat{U}\str\pi_N(\alg{\hilb})''\widehat{U}\subset \pi_N(\alg{\hilb})''$
by~(\ref{UImpl}). Hence, using~(\ref{QSpectrDec}), we see that
%
\begin{align*}
(\Omega_N, \widehat{U}\str\ep{\iu\lambda\widehat{Q}}\widehat{U}\ep{-\iu\lambda\widehat{Q}}\,\Omega_N)&=\sum_{n\in\mathbb{Z}}\,(\Omega_N,
P_n\widehat{U}\str\ep{\iu\lambda\widehat{Q}}\widehat{U}P_n\,\Omega_N)\ep{-\iu\lambda n} \\
&=\sum_{n,m\in\mathbb{Z}}\,(\Omega_N,P_n\widehat{U}\str P_m\widehat{U}P_n\,\Omega_N)\ep{\iu\lambda (m-n)}
\end{align*}
%
is of the stated form.\teop\newline

\subsection{Part (i)}

Since the projection $Q$ commutes with $N$, see (\ref{QN}), we have
%
\begin{align*}
\ep{\iu\lambda NQ}&=\Id+(\ep{\iu\lambda N}-\Id)Q\,, \\
\ep{-\iu\lambda N_UQ_U}&=\Id+Q_U(\ep{-\iu\lambda N_U}-\Id)\,.
\end{align*}
%
We insert these equations in the definition (\ref{defnicedet}) of
$\widetilde{D}(\lambda)$. Moreover,
%
\begin{equation}
N_U-N=N^{1/2}(N_U^{1/2}-N^{1/2})+(N_U^{1/2}-N^{1/2})N_U^{1/2}\in\ideal{1}\,,
\label{deltaN}
\end{equation}
%
so that
%
\begin{equation*}
\ep{-\iu\lambda N}-\ep{-\iu\lambda N_U}=
\iu\int_0^\lambda\ep{-\iu(\lambda-s)N_U}(N_U-N)\ep{-\iu s N}\,\dd s
\end{equation*}
%
also belongs to the trace class ideal. Rather than proving
$\widetilde{D}(\lambda)\in\Id+\ideal{1}$ for
$\widetilde{D}(\lambda)$ we may thus do so for the expression
%
\begin{multline*}
[\Id+Q_U(\ep{-\iu\lambda N}-\Id)]N'[\Id+(\ep{\iu\lambda N}-\Id)Q]+
[\Id+Q_U(\ep{\iu\lambda N'}-\Id)]N[\Id+(\ep{-\iu\lambda N'}-\Id)Q]\\
\begin{aligned}
=&N'+N&&+Q_U[(\ep{-\iu\lambda N}-\Id)N'+(\ep{\iu\lambda N'}-\Id)N]+[N'(\ep{\iu\lambda N}-\Id)+N(\ep{-\iu\lambda N'}-\Id)]Q \\
&&&+Q_U[(\ep{-\iu\lambda N}-\Id)N'(\ep{\iu\lambda N}-\Id)+(\ep{\iu\lambda N'}-\Id)N(\ep{-\iu\lambda N'}-\Id)]Q \\
=&\Id&&+Q_U\left[(\cos(\lambda N)-\Id)N'+(\cos(\lambda N')-\Id)N-\iu\sin(\lambda N) N'+\iu\sin(\lambda N') N\right] \\
&&&+Q\left[(\cos(\lambda N)-\Id)N'+(\cos(\lambda N')-\Id)N+\iu\sin(\lambda N) N'-\iu\sin(\lambda N') N\right] \\
&&&+2Q_UQ\left[(\Id-\cos(\lambda N))N'+(\Id-\cos(\lambda N'))N\right] \\
=&\Id&&+(Q_U^2+Q^2-2Q_UQ)[(\cos(\lambda N)-\Id)N'+(\cos(\lambda N')-\Id)N] \\
&&&+\iu(Q-Q_U)[\sin(\lambda N)N'-\sin(\lambda N')N]\,.
\end{aligned}
\end{multline*}
%
With the help of the functions $f(x)=(\cos x-1)/x$ and
$g(x)=(\sin x)/x$, which are bounded also at $x=0$, the expression is
rewritten as 
%
\begin{equation*}
\Id+[(Q-Q_U)Q-Q_U(Q-Q_U)]NN'\lambda(f(\lambda N)+f(\lambda N'))+
\iu(Q-Q_U)NN'\lambda(g(\lambda N)-g(\lambda N'))\,.
\end{equation*}
%
Besides of $Q\sqrt{NN'}\in\ideal{1}$, see eq.~(\ref{mixedcond}),
we have $Q_U\sqrt{NN'}=U\str
Q\sqrt{N_{U\str}N'_{U\str}}U\in\ideal{1}$ by eq.~(\ref{NTrCl}),
and hence $(Q-Q_U)\sqrt{NN'}\in\ideal{1}$, cf.~(\ref{weaker}).
This makes the claim manifest.
\teop\newline

In the zero temperature case, where $N$ is
a projection, the above proof simplifies considerably due to $NN'=0$.

\subsection{Preliminaries}\label{pre}

We recall a few results about Bogoliubov transformations, first inner and then
others.

Given a bounded operator $A$ on $\hilb$, operators $\Gamma(A)$ and
$\dd\Gamma(A)$ are usually defined on the Fock space over $\hilb$. Following
\cite{ArakiWyss} we
define them instead as elements of the CAR-algebra $\alg{\hilb}$, when $A$ is
of finite rank.

\begin{itemize}
\item For rank one operators $A_i=|f_i\rangle\langle g_i|$, $(i=1,\ldots,n)$,
we set
\begin{equation}
\label{DefDGamma}
\dd\Gamma(A_1,\,\ldots,\,A_n)=
a\str(f_n)\cdots a\str(f_1)\,a(g_1)\cdots a(g_n)\,.
\end{equation}
The definition is extended by multilinearity to operators $A_i$ of finite
rank. The result is independent of the particular decomposition into rank
one operators.
\item For $U-\Id$ of finite rank, we set
\begin{equation*}
\Gamma(U)=\sum_{n=0}^\infty\frac{1}{n!}\dd\Gamma(\underbrace{U-\Id,\,\ldots,\, U-\Id}_{n})\,,
\end{equation*}
where the term $n=0$, in which no arguments are present, is read as
$\dd\Gamma=1$. The sum is finite, because the terms with
$k>\mathrm{rank}(U-1)$ vanish.
\end{itemize}
%
The elements of $\alg{\hilb}$ just defined share the properties of the
operators on Fock space known by the same notation.

\begin{lma}
Let $U-1$ be of finite rank. Then
%
\begin{align}
\Gamma(U)a\str(f)&=a\str(Uf)\Gamma(U)\,, \label{prp1}\\
\Gamma(U_1U_2)&=\Gamma(U_1)\Gamma(U_2)\,.\label{prp3}
\end{align}
%
In particular, $\Gamma(U)$ is unitary if $U$ is.
\end{lma}
%
\paragraph{Proof.} We have
%
\begin{equation}
\label{dga}
\dd\Gamma(A_1,\ldots,A_n)a\str(f)=a\str(f)\dd\Gamma(A_1,\ldots,A_n)+\sum_{i=1}^{n}\,a\str(A_if)\dd\Gamma(A_1,\ldots,\widehat{A}_i,\ldots,A_n)\,
\end{equation}
where the hat indicates omission. In the rank one case,
$A_i=|f_i\rangle\langle g_i|$, this follows
from~(\ref{DefDGamma}) and from $(g_i,f)a\str(f_i)=a\str(A_i f)$. In the
general case, by multilinearity. Thus,
%
\begin{align*}
\Gamma(U)a\str(f)&=a\str(f)\Gamma(U)+a\str((U-1)f)\sum_{n=1}^{\infty}\frac{1}{(n-1)!}\,\dd\Gamma(U-1,\ldots,U-1) \\
&=a\str(f)\Gamma(U)+a\str((U-1)f)\Gamma(U) = a\str(Uf)\Gamma(U)\,,
\end{align*}
%
since we applied (\ref{dga}) with $n$ equal entries $A_i=U-1$.

\noindent We have
%
\begin{align*}
&\dd\Gamma(A_1,\ldots,A_n)\dd\Gamma(B_1,\ldots,B_m)= \\
&=\sum_{l=0}^{\min(n,m)}\sum_{\mathcal{C}_l}\,\dd\Gamma(A_{i_1}B_{j_1},\ldots,A_{i_l}B_{j_l},A_1,\ldots,\widehat{A}_{i_s},\ldots,A_n,B_1,\ldots,\widehat{B}_{j_s},\ldots,B_m)\,,
\end{align*}
%
where the second sum runs over all $l$-contractions $(i_1, j_1),\ldots,(i_l, j_l)$ with $i_1<\ldots<i_l,j_{i_s}\neq j_{i_r}$. In the rank one
case, which implies the general one, this is just Wick's lemma for normal
ordered products. Thus
%
\begin{align*}
\Gamma(U_1)&\Gamma(U_2)=\sum_{n=0}^{\infty}\frac{1}{n!}\,\dd\Gamma(U_1-1,\ldots,U_1-1)\cdot\sum_{m=0}^{\infty}\frac{1}{m!}\,\dd\Gamma(U_2-1,\ldots,U_2-1)
\\
&=\sum_{n,m=0}^{\infty}\sum_{l=0}^{\min(n,m)}\frac{1}{l!(n-l)!(m-l)!}\,\dd\Gamma((U_1-1)(U_2-1),\ldots,U_1-1,\ldots,U_2-1,\ldots)
\end{align*}
%
with entries repeated $l$, $n-l$, $m-l$ times. In fact, the number of $l$-contractions is
\begin{equation*}
\frac{1}{l!}\frac{n!}{(n-l)!}\frac{m!}{(m-l)!}\,.
\end{equation*}
%
Setting $n-l=:s$, $m-l=:t$, $l+s+t=:r$, we have
%
\begin{align*}
\Gamma(U_1)\Gamma(U_2)&=\sum_{r=0}^{\infty}\sum_{\begin{subarray}{c} l,s,t \\
l+s+t=r\end{subarray}}\frac{1}{l!\,s!\,t!}\,\dd\Gamma((U_1-1)(U_2-1),\ldots,U_1-1,\ldots,U_2-1,\ldots) \\
&=\sum_{r=0}^{\infty}\frac{1}{r!}\,\dd\Gamma((U_1-1)(U_2-1)+(U_1-1)+(U_2-1),\ldots)
\end{align*}
%
since there are ${r!}/l!\,s!\,t!$ ways to pick terms from each entry of
the last line. Since $(U_1-1)(U_2-1)+(U_1-1)+(U_2-1)=U_1U_2-1$,
the proof is complete.\teop\newline
%

If $O$ is an operator on $\hilb$ such that $O-\Id$ is in the trace
class, its {\rm Fredholm determinant} is defined by
\begin{equation}
\label{FredDet}
\det O=\sum_{k=0}^{\infty}\tr\wedge^k(O-\Id)\,.
\end{equation}
This extends the usual definition of the determinant in the finite
dimensional case.
%
\begin{lma}
\label{scalprod}
Let $A$ be a finite rank operator, and $0\leq N\leq\Id$. Then
\begin{equation*}
\omega_N(\dd\Gamma(\underbrace{A,\,\ldots,\,A}_k))=\tr\wedge^k(AN)\,.
\end{equation*}
Moreover, if $U$ is such that $U-\Id$ is of finite rank, then
\begin{equation}
\omega_N(\Gamma(U))=\det((\Id-N)+UN)\,.
\label{det}
\end{equation}
\end{lma}
\paragraph{Proof.} The trace of a finite rank operator
$A=\sum_{i=1}^m f_i(g_i, \cdot)$ is $\tr A=\sum_{i=1}^m (g_i,f_i)$. By the same
token, that of
%
\begin{equation*}
\wedge^kA=\sum_{i_1,\ldots i_k=1}^m \frac{1}{k!}\sum_{\sigma\in S_k}
(-1)^\sigma\otimes_{\alpha=1}^kf_{i_{\sigma(\alpha)}}(g_{i_\alpha},  \cdot)
\end{equation*}
%
is
%
\begin{equation*}
\tr\wedge^kA=
\sum_{1\leq i_{1}<\ldots<i_{k}\leq m}
\det(g_{i_{\alpha}}, f_{i_{\beta}})_{ \alpha, \beta=1}^k\,.
\end{equation*}
%
Since the $a^{\#}(f)$ anticommute, we have
%
\begin{equation*}
\dd\Gamma(A,\,\ldots,\,A)=
\sum_{1\leq i_{1}<\ldots<i_{k}\leq m}a\str(f_{i_1})\cdots a\str(f_{i_k})\,a(g_{i_k})\cdots a(g_{i_1})\,,
\end{equation*}
%
whose expectation value is computed by (\ref{state}) as
%
\begin{equation*}
\omega_N(\dd\Gamma(A,\,\ldots,\,A))
=\sum_{1\leq i_{1}<\ldots<i_{k}\leq m}
\det(Ng_{i_{\alpha}}, f_{i_{\beta}})_{ \alpha, \beta=1}^k =
\tr\wedge^k(AN) \,,
\end{equation*}
because the decomposition of $AN$ differs from that of $A$ by
$Ng_i$ in place of $g_i$. This proves the first part of the lemma. The second
part is now an application of the definition of the determinant,
eq.~(\ref{FredDet}). Indeed,
$\omega_N(\Gamma(U))=1+\sum_{k=1}^{\infty}\tr\wedge^k((U-\Id)N)=\det(\Id+(U-\Id)N)$.\teop\newline
%, and we note that the determinant is well-defined.

We recall a few results on Bogoliubov
transformations. Their proofs can be found \eg in \cite{Araki_2},
where however CAR-algebras are introduced in the self-dual guise.
A remark at the end of this subsection is intended as an aid to
translation. The first result is the Shale-Stinespring 
criterion \cite{ShSt} about unitary implementability (see \eg \cite{Lundberg},
\cite{Araki_2} Thm. 6.3 (1)).
\begin{prop}
\label{BogImpl_1} Let $P$ be a projection and $V$ a unitary
operator on $\hilb$. The Bogoliubov automorphism induced by $V$ on
$\hilb$, \ie $a(f)\mapsto a(Vf)$, is unitarily implementable in
the representation $\pi_P$ if and only if $PV(\Id-P)$ and
$(\Id-P)VP$ are in the Hilbert-Schmidt class, $\ideal{2}$.
\end{prop}
In particular, an equivalent condition is
$[P, V]=PV(\Id-P)-(\Id-P)VP\in\ideal{2}$. There is a version of this
proposition for groups (\cite{Araki_2}, Thm. 6.10 (2, 3)).
\begin{prop}
\label{BogImpl_2}
Let $V$ in the previous proposition be replaced by a 1-parameter unitary group
$V_\lambda$, ($\lambda\in\mathbb{R}$), such that $V_\lambda$ is norm
continuous and $PV_\lambda(\Id-P)$ is continuous in the
$\ideal{2}$-norm. Then $V_\lambda$ has an implementer of the form
$\widehat{V}_\lambda=\exp(\iu\lambda \widehat{v})$, where $\widehat{v}$ is a
self-adjoint operator on $\hilb_{P}$. The requirement
%
\begin{equation}
\label{nrmlz}
(\Omega_P,\widehat{v}\Omega_P)=0
\end{equation}
%
may be imposed, in which case $\widehat{v}$ is unique.
\end{prop}
An equivalent condition,
is $V_\lambda=\exp(\iu\lambda v)$ with $v$ a bounded, selfadjoint
operator on $\hilb$, and $Pv(\Id-P)\in\ideal{2}$.

The next result is about the
continuity of the implementation, see \cite{Araki_2}, Thm.~6.10 (7).
\begin{prop}
\label{continuity} Let $v$ and $v_n$, ($n=0,1,\ldots$), satisfy the hypotheses
of Prop.~\ref{BogImpl_2}, and let $\widehat{v}$, $\widehat{v_n}$ satisfy the
normalization (\ref{nrmlz}). If $(v_n)$ converge strongly to $v$ and if
$\| P(v_n-v)(\Id-P)\|_2\rightarrow 0$ as $n\to\infty$, then
\begin{equation*}
\mathrm{s-}\lim_{n}\ep{\iu\lambda \widehat{v_n}}=\ep{\iu\lambda \widehat{v}}\,.
\end{equation*}
\end{prop}

The last preliminary is concerned with the twisted duality of CAR-algebras.
Let $P$ be an orthogonal projection on $\hilb$,
$\mathcal{K}\subset\hilb$ a closed subspace, and
$\mathcal{K}^{\perp}$ its orthogonal complement. Let
$\widetilde{\mathcal{A}}({\mathcal{K}^\perp})$ be the von Neumann algebra
generated by $\widehat{\Lambda}\pi_P(a(f))$, ($f\in\mathcal{K}^\perp$), where
$\widehat{\Lambda}$ is the parity. Then \cite{HaagDuality, CARDuality}
%
\begin{equation}
\label{duality}
\pi_P(\alg{K})'=\widetilde{\mathcal{A}}({\mathcal{K}^{\perp}})\,.
\end{equation}
%
The implementers $\widehat{V}$ from Prop.~\ref{BogImpl_1} commute with
parity:
Let $\widehat{\Lambda}:\:\hilb_P\rightarrow\hilb_P$ be the unitary
implementation of the *-automorphism $a(f)\mapsto a(-f)$ which is uniquely
determined by $\widehat{\Lambda}\Omega_P=\Omega_P$. Then
%
\begin{equation}
[\widehat{V},\widehat{\Lambda}]=0\,,
\label{comm}
\end{equation}
%
see \cite{Araki_2}, Thm. 6.3 (3), Thm. 6.7 (2). We will actually apply this
fact only to $\widehat{V}=\widehat{V}_\lambda$ as in Prop.~\ref{BogImpl_2}, in
which case it can be verified as follows. Since $[V_\lambda, -1]=0$, the
operators $\widehat{\Lambda}\widehat{V}_\lambda$ and
$\widehat{V}_\lambda\widehat{\Lambda}$ implement the same Bogoliubov
automorphism, whence
$\widehat{\Lambda}\widehat{V}_\lambda=
c_\lambda\widehat{V}_\lambda\widehat{\Lambda}$ with $|c_\lambda|=1$. From
$(\Omega_P, \widehat{V}_\lambda\Omega_P)=
(\widehat{\Lambda}\Omega_P, \widehat{\Lambda}\widehat{V}_\lambda\Omega_P)
=c_\lambda(\Omega_P, \widehat{V}_\lambda\Omega_P)$ we find $c_\lambda=1$
for small $\lambda$, because $(\Omega_P, \widehat{V}_\lambda\Omega_P)\to 1$,
($\lambda\to 0$). The conclusion extends to all $\lambda$ by the group
property.\\

\noindent
{\bf Remark.} In order to make contact with the repeatedly cited
article \cite{Araki_2} we recall that a self-dual CAR-algebra
$\alg{\widetilde\hilb,\Gamma}$ is given in terms of a separable Hilbert space
$\widetilde\hilb$ equipped with a conjugation $\Gamma$. Its
generators $B(h)$ are linear in $h\in\widetilde\hilb$ and the relations are
$B(h)^*=B(\Gamma h)$ and $\{B(h),B(h')^*\}=(h',h)1$.
A projection $\widetilde P$ on $\widetilde\hilb$ satisfying
$\widetilde P+\Gamma\widetilde P\Gamma=1$ defines a pure state
$\omega$ on the algebra through
%
\[
\omega(B(h)^*B(h))=0\,, \qquad (\widetilde Ph=0)\,.
\]
%
The algebra $\alg{\hilb}$ is connected to the above by picking a
conjugation $C$ on $\hilb$ and by setting
%
\[
\widetilde\hilb=\hilb\oplus \hilb\,,\qquad
\Gamma (f\oplus g)=Cg\oplus Cf\,,\qquad
B(f\oplus g)=a^*(f)+a(Cg)\,.
\]
%
States defined by $P$ and $\widetilde P$ then agree if
$\widetilde P(f\oplus g)=(1-P)f\oplus CPCg$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Parts (ii-iv)}

{\bf Pure state:} $N=N^2$

Existence: In the case $V_\lambda=\exp(\iu\lambda Q)$ we have
$[N, V_\lambda]=0$ by eq.~(\ref{QN}), so that existence of a unitary
implementer $\widehat{V}_\lambda=\exp(\iu\lambda \widehat{Q})$ is trivial by
Prop.~\ref{BogImpl_2}. Similarly, in the case $V=U$ we have
$[N,U]=(N-N_{U\str})U\in\ideal{2}$ by eq.~(\ref{deltaN}). Hence it is also
implementable by Prop.~\ref{BogImpl_1}.

Uniqueness:
Let $\widehat{V}$ denote either of $\exp(\iu\lambda \widehat{Q})$ or
$\widehat{U}$. Suppose $\widehat{V}_1$ and $\widehat{V}_2$ both implement the
same transformation. Then
$\widehat{V}_1\widehat{V}_2\str\pi_N(a(Vf))=
\pi_N(a(Vf))\widehat{V}_1\widehat{V}_2\str$.
Thus $\widehat{V}_1=(\widehat{V}_1\widehat{V}_2\str)\widehat{V}_2$ and
$\widehat{V}_2$ differ by left multiplication with
$\widehat{V}_1\widehat{V}_2\str\in \pi_N(\alg{\hilb})'$. In the pure case the
cyclic representation is irreducible, whence $\widehat{U}$ is unique up to a
phase and $\widehat{Q}$ up to an additive constant. As mentioned in Remark 3,
property (iii) is empty in this case.\\

\noindent{\bf Mixed state:} $0<N<\Id$

Given $0\leq N\leq\Id$ on $\hilb$, we consider its {\it purification}
\begin{equation*}
P_N=\left(\begin{array}{cc} N & \sqrt{NN'} \\ \sqrt{NN'} &
    N'\end{array}\right)
=P_N^2
\end{equation*}
on $\hilb\oplus\hilb$, together with the cyclic representation
$(\hilb_{P_N}, \pi_{P_N}, \Omega_{P_N})$ of the state defined by $P_N$ on $\alg{\hilb\oplus\hilb}$. We
can identify
\begin{equation*}
\alg{\hilb}\cong\alg{\hilb\oplus0}
\end{equation*}
via $a(f)=a(f\oplus0)$, and
%
\begin{equation}
\label{equiv}
\hilb_N\equiv\pi_{P_N}(\alg{\hilb\oplus0})''\Omega_{P_N}\subset\hilb_{P_N}\,,
\qquad
\Omega_N\equiv\Omega_{P_N}\,,\qquad
\pi_N(a)\equiv\pi_{P_N}(a)\upharpoonright\hilb_N\,,
\end{equation}
%
since these objects satisfy
%
\begin{equation}
(\Omega_{P_N},\pi_{P_N}(a\str(f\oplus0)a(g\oplus0))\Omega_{P_N})
=(g\oplus0, P_N(f\oplus0))=(g, Nf)\,,
\end{equation}
%
as required by (\ref{GNS}). We can not handle the most general mixed case
$0\le N\le 1$. The reason comes from the following
lemma, whose proof is postponed till the end of the section.

\begin{lma}\label{cyclic}
Assume $0<N<1$ (strict inequality). Then $\,\Omega_{P_N}$ is cyclic in
$\hilb_{P_N}$ for $\pi_{P_N}(\alg{\hilb\oplus0})$. In particular, we have
equality in~(\ref{equiv}), $\hilb_N=\hilb_{P_N}$.
\end{lma}


 A unitary $V$ on $\hilb$ induces two automorphisms on
$\alg{\hilb\oplus\hilb}$: (a) $a(f\oplus g)\mapsto a(Vf\oplus g)$,
and (b) $a(f\oplus g)\mapsto a(Vf\oplus Vg)$, whose implementation
may be envisaged:
%
\begin{align}
\text{(a)}\qquad
\widehat{V}\pi_{P_N}(a(f\oplus g))\widehat{V}\str
=&\pi_{P_N}(a(Vf\oplus g))\,,\label{impla} \\
\text{(b)}\qquad
\widehat{V}\pi_{P_N}(a(f\oplus g))\widehat{V}\str
=&\pi_{P_N}(a(Vf\oplus Vg))\,. \nonumber
\end{align}
%
Both choices for $\widehat{V}$ would provide an implementation for $V$ in
the representation $\pi_N$ on
$\hilb_N$. Only in the first case we have
$\widehat{V}\in\pi_N(\alg{\hilb})''$. Indeed, by (\ref{duality}) applied to
$\hilb\oplus 0\subset \hilb\oplus\hilb$ instead of $\mathcal{K}\subset\hilb$,
we need to check $\widehat{V}\in \widetilde{\mathcal{A}}(0\oplus\hilb)'$.
This however follows from $[\widehat{V}, \pi_{P_N}(a(0\oplus g))]=0$, see
(\ref{impla}), and from $[\widehat{V},\widehat{\Lambda}]=0$, see
(\ref{comm}).

In order to determine the existence of these implementations we compute
\begin{equation*}
\left[P_N, \left(\begin{array}{cc} U_1 & 0 \\ 0 & U_2 \end{array}\right)\right]=\left(\begin{array}{cc} [N,U_1] & \sqrt{NN'}\,U_2-U_1\sqrt{NN'} \\ \sqrt{NN'}\,U_1-U_2\sqrt{NN'} & [N',U_2] \end{array}\right)\,,
\end{equation*}
and see that that of (a) (\ie $U_1=V$ and $U_2=\Id$) is granted if
\begin{equation}
\label{condcase1}
[N,V]\in\ideal{2}\,,\qquad(\Id-V)\sqrt{NN'}\in\ideal{2}\,;
\end{equation}
and that of (b) ($U_1=V=U_2$) if
\begin{equation}
\label{condcase2}
[N,V]\in\ideal{2}\,,\qquad[\sqrt{NN'}, V]\in\ideal{2}\,.
\end{equation}

We can now complete the proof of parts (ii) and (iii) of the
theorem. For $V=\exp(\iu\lambda Q)$, we use (a). Then eqs.~(\ref{condcase1})
hold true by eqs.~(\ref{QN}, \ref{mixedcond}):
$(\Id-\ep{\iu\lambda Q})\sqrt{NN'}=(1-\ep{\iu\lambda})Q\sqrt{NN'}$.
This also proves (iii). For $V=U$, we use (b), with
conditions~(\ref{condcase2}) holding by eq.~(\ref{NTrCl}, \ref{deltaN}).

Part~(iv) is readily proven as follows. Like in the pure case, the
implementers are unique up to left multiplication by an element of
$\pi_N(\alg{\hilb})'$ (which is larger than the multiples of the identity
since the representation is reducible).
Thus $\exp{(\iu\lambda\widehat{Q})}\in\pi_N(\alg{\hilb})''$ still implies its
uniqueness up to a phase because of (\ref{factor}).\teop\\

\noindent
{\bf Remark.} If it were for property (ii) only, one could adopt method (b)
also for $V=\exp(\iu\lambda Q)$.

\paragraph{Proof of Lemma~\ref{cyclic}.} The space $\hilb_{P_N}$ is spanned by the vectors
%
\begin{equation}
\label{H_PN}
\prod_{i=1}^n\,\pi_{P_N}(a^{\#}(f_i\oplus g_i))\Omega_{P_N}\,.
\end{equation}
%
It suffices to show that they can be approximated arbitrarily well by a sum of
such vectors where, however, $g_i=0$. To this end, we first note that
%
\begin{align*}
\pi_{P_N}(a\str(P_N(f\oplus g)))\Omega_{P_N}=0\,, \\
\pi_{P_N}(a(P_N'(f\oplus g)))\Omega_{P_N}=0\,,
\end{align*}
%
where $P_N'=1-P_N$. This follows from~(\ref{state}) for $P_N$ instead of $N$, and implies in turn
%
\begin{align*}
\| \pi_{P_N}(a\str(f\oplus g))\Omega_{P_N}\|\leq\|P_N'(f\oplus g)\|\,, \\
\|\pi_{P_N}(a(f\oplus g))\Omega_{P_N}\|\leq\|P_N(f\oplus g)\|\,.
\end{align*}
%
Let us first consider the case where the last factor in~(\ref{H_PN}) is an annihilation operator and set $f_n\oplus g_n=:f\oplus g$. We have
%
\begin{equation*}
P_N(f\oplus g-\widetilde{f}\oplus 0)=\left(\begin{array}{c}
\sqrt{N} \\ \sqrt{N'}
\end{array}\right) \bigl(\sqrt{N}(f-\widetilde{f})+\sqrt{N'}g\bigr)\,.
\end{equation*}
%
A vector $\widetilde{f}\in\hilb$ is well-defined by
%
\begin{equation*}
\sqrt{N}\widetilde{f}:=\sqrt{N} f+\sqrt{N'}F(N\geq\epsilon)g\,,
\end{equation*}
%
where $F(N\geq\epsilon)$ is the spectral projection for $N$ on $[\epsilon,
\infty)$ and $\epsilon>0$. Thus,
%
\begin{equation*}
\|P_N(f\oplus g-\widetilde{f}\oplus 0)\|\leq\|F(N < \epsilon)g\|
\end{equation*}
%
can be made arbitrarily small because of $\Ker{N}=\{0\}$. If the last factor is a creation operator, the arguments proceed similarly using $\Ker{N'}=\{0\}$. Hence the announced replacement can be performed in the last factor. After anticommuting it to the left, the claim is reduced to products with fewer factors, for which it holds by induction.\teop

\subsection{Part (v)}

The idea of the proof is to approximate the Bogoliubov automorphism induced by
$\ep{\iu\lambda Q}$ by means of inner automorphisms, as introduced in
Subsection~\ref{pre}. The generating function on the l.h.s. of 
(\ref{Main}) then becomes computable by Lemma~\ref{scalprod}.
We present separate proofs in the pure and the mixed case. The
second proof, while applying to both cases, is longer than the one we
give for pure states. Both depend on Prop.~\ref{continuity}.

{\bf Pure state.}
Let $F$ be a finite rank operator on $\hilb$ with $[F,N]=0$. As such, it has
an implementation in the cyclic representation $\pi_N$; its non-uniqueness
does not affect the l.h.s. of
%
\begin{align}
(\Omega_N,
\widehat{U}\str\ep{\iu\lambda\widehat{F}}\widehat{U}\ep{-\iu\lambda\widehat{F}}
\Omega_N)
&=(\Omega_N,
\widehat{U}\str\pi_N(\Gamma(\ep{\iu\lambda F}))\widehat{U}
\pi_N(\Gamma(\ep{-\iu\lambda F}))
\Omega_N) \nonumber\\
\label{approxscal}
&=(\Omega_N, \pi_N(\Gamma(U\str\ep{\iu\lambda F}U\ep{-\iu\lambda F}))\Omega_N)\,;
\end{align}
on the r.h.s. we used that $\pi_N(\Gamma(\ep{\iu\lambda F}))$ is one possible
implementation of $\ep{\iu\lambda F}$ by (\ref{prp1}) with
$\ep{\iu\lambda F}$ in place of $U$; the second
line follows by (\ref{UImpl}), which implies
$\widehat{U}\str\pi_N(\Gamma(\ep{\iu\lambda F}))\widehat{U}=
\pi_N(\Gamma(U\str\ep{\iu\lambda F}U))$, and by (\ref{prp3}). Another choice
for $\widehat{F}$ is fixed by
%
\begin{equation}
(\Omega_N,\widehat{F}\Omega_N)=0\,,
\label{nrmlz1}
\end{equation}
%
and we may ask the same normalization for $\widehat{Q}$.
%
\begin{lma}
\label{Fnpure}
There is a sequence of finite dimensional orthogonal projections $F_n$ such
that
%
\begin{equation}
[F_n,N]=0\,,\qquad\mathrm{s-}\lim_{n}\,F_n=Q\,.
\label{FnN}
\end{equation}
\end{lma}
%
%%%%%%%%%%%%%%%%%%
\paragraph{Proof.} We note that $(NQ)^2=NQ$, so that $Q=NQ+N'Q$ is an
orthogonal splitting of $Q$. Let $F_n=F_n^{(1)}+F_n^{(2)}$, where $F_n^{(1)}$,
resp. $F_n^{(2)}$, is a subprojection of $NQ$ (\ie $F_n^{(1)}NQ=F_n^{(1)}$),
resp. of $N'Q$, with $F_n^{(1)}\stackrel{s}{\to} NQ$, and $F_n^{(2)}\stackrel{s}{\to} N'Q$. Clearly, $F_n\stackrel{s}{\to}Q$ and
\begin{equation*}
[F_n^{(1)},N]=[N',F_n^{(1)}]=N'F_n^{(1)}-F_n^{(1)}N'=
N'NQF_n^{(1)}-F_n^{(1)}QNN'=0\,,
\end{equation*}
since $NN'=0$. The same holds for $F_n^{(2)}$, and thus for $F_n$.\teop\newline

By (\ref{FnN}, \ref{nrmlz1}) the assumptions of Prop.~\ref{continuity} are
satisfied for the sequence $(F_n)$ and its limit $Q$. Therefore,
\begin{equation}
\label{scalprodconv}
(\Omega_N, \widehat{U}\str\ep{\iu\lambda\widehat{Q}}\widehat{U}\ep{-\iu\lambda\widehat{Q}} \Omega_N)=\lim_{n\to\infty}(\Omega_N, \widehat{U}\str\ep{\iu\lambda\widehat{F_n}}\widehat{U}\ep{-\iu\lambda\widehat{F_n}} \Omega_N)\,.
\end{equation}
By eq.~(\ref{approxscal}, \ref{det}, \ref{FnN}) the inner product on the
r.h.s. equals
\begin{equation*}
\det(N'+\ep{\iu\lambda U\str F_n U}\ep{-\iu\lambda F_n}N)=
\det(\ep{-\iu\lambda N_U F_{nU}}N'\ep{\iu\lambda N F_{n}}+
\ep{\iu\lambda N'_U F_{nU}}N\ep{-\iu\lambda N' F_n})\,,
\end{equation*}
where we multiplied the determinant by
\begin{equation}
\label{purecasemultdet}
1=\det(\ep{-\iu\lambda N_U F_{nU}})\cdot\det(\ep{\iu\lambda N F_{n}})\,,
\end{equation}
like in the heuristic derivation (\ref{formal}); but unlike there, this step
is now correct, since $F_n$ is of finite rank. We also used $[F_n,N]=0$.
Finally, we claim that the operator under the last determinant converges to
\begin{equation}
\label{needalabel}
\ep{-\iu\lambda N_U Q_U}N'\ep{\iu\lambda N Q}+\ep{\iu\lambda N'_U Q_U}N\ep{-\iu\lambda N' Q}=\ep{-\iu\lambda N_U Q_U}N'+\ep{\iu\lambda N'_U Q_U}N
\end{equation}
in trace class norm, \ie the same expression with $Q$ in place of $F_n$.
The r.h.s. is obtained using $\exp(\iu\lambda
NQ)=\Id+NQ(\exp(\iu\lambda)-1)$ and $NN'=0$. The convergence implies that of
the determinants: Indeed, for $A-\Id$, $B-\Id\in\ideal{1}$, we have (\cite{RSIV}, Lemma~XIII.17.1 (d))
\begin{equation*}
|\det A-\det B|\leq\|A-B\|_1\ep{(\|A-\Id\|_1+\|B-\Id\|_1+1)}\,.
\end{equation*}
Upon conjugating with $U$, it is enough to show
\begin{equation*}
\|(\ep{-\iu\lambda N F_n}-\ep{-\iu\lambda N Q})N'_{U\str}\|_1
\longrightarrow 0\,,
\end{equation*}
and similarly with $N$ and $N'$ interchanged. This operator equals
$\ep{-\iu\lambda}-1$ times
%
\begin{equation*}
N(F_n-Q)N'_{U\str}=
(F_n-Q)NN'+(F_n-Q)N(N'_{U\str}-N')\,.
\end{equation*}
%
The first term vanishes, and the second tends to $0$ in the trace class norm
as $n\to\infty$, because of
%
\begin{equation}\label{trclconv}
X_n\stackrel{s}{\longrightarrow}0\,,\quad Y\in\ideal{1}\quad\Longrightarrow\quad\| X_nY \|_1\longrightarrow 0\;.
\end{equation}
%
\teop\newline

{\bf Mixed state.}
Let us start by proving a result analogous to Lemma~\ref{Fnpure}:
%
\begin{lma}
\label{lmahyp0}
Let $P$, $Q$ be orthogonal projections in a separable Hilbert space $\hilb$ with
\begin{equation}
\label{lmahyp}
[Q,P]\in\ideal{1}\,.
\end{equation}
Then there are finite dimensional subprojections $F_n$ of $Q$ with
\begin{equation}
\|[F_n-Q,P]\|_1\longrightarrow 0\,,\quad(n\to\infty)\,.
\end{equation}
\end{lma}

\paragraph{Proof.} We split $Q$ as
%
\begin{equation}
\label{splitting}
Q=QPQ+Q(1-P)Q\equiv L_1+L_0\,,
\end{equation}
%
and observe that $[Q,L_1]$=0 and
\begin{gather}
\label{PLone}
(P-1)L_1\in\ideal{1}\,, \\
L_1^2-L_1=QP[Q,P]Q\in\ideal{1}\,.\nonumber
\end{gather}
%
By the last property, the only possible accumulation points in the spectrum of $L_1$ are $0$ and $1$. In particular, there is an $x\in(0,1)$ which is not in the spectrum. Let $Q_1$ be the spectral projection of $L_1$ associated with $(x,\infty)$. It may be represented as
%
\begin{equation*}
Q_1=\frac{1}{2\pi\iu}\oint_{\mathcal{C}}(z-L_1)^{-1}dz\,,
\end{equation*}
%
where $\mathcal{C}\subset\mathbb{C}$ is a contour encircling that part of the spectrum only. Using $\oint_{\mathcal{C}}z^{-1}dz=0$, due to $x>0$, we have
%
\begin{align}
(P-1)Q_1&=\frac{1}{2\pi\iu}\oint_{\mathcal{C}}(P-1)\left((z-L_1)^{-1}-z^{-1}\right)dz \nonumber \\
\label{PQone}
&=\frac{1}{2\pi\iu}\oint_{\mathcal{C}}(P-1)L_1(z-L_1)^{-1}z^{-1}dz\in\ideal{1}
\end{align}
%
by~(\ref{PLone}). On the subspace $\Ran Q$, the projection $Q_1$, defined in terms of $L_1$ and $x$ is complementary to the one, $Q_0$, similarly defined by $L_0$ and $1-x$, see~(\ref{splitting}). Since $1-x>0$, we have
%
\begin{equation}
\label{PQnought}
PQ_0\in\ideal{1}
\end{equation}
%
by analogy to~(\ref{PQone}). Let now $F_n^{(i)}$, $(i=0,1)$, be a sequence of finite dimensional subprojections of $Q_i$ with $F_n^{(i)}\stackrel{s}{\rightarrow}Q_i$. Then
%
\begin{align*}
[F_n^{(0)}-Q_0,P]&=(F_n^{(0)}-Q_0)P-P(F_n^{(0)}-Q_0)=(F_n^{(0)}-Q_0)Q_0P-PQ_0(F_n^{(0)}-Q_0)\,, \\
[F_n^{(1)}-Q_1,P]&=[F_n^{(1)}-Q_1,P-1]=(F_n^{(1)}-Q_1)Q_1(P-1)-(P-1)Q_1(F_n^{(1)}-Q_1)\,,
\end{align*}
%
are trace class by~(\ref{PQone}, \ref{PQnought}), and converge to zero in the corresponding norm by~(\ref{trclconv})
and $\| T\str \|_1=\| T \|_1$. Thus $F_n=F_n^{(0)}+F_n^{(1)}$ is seen to have the stated properties.\teop\newline

We apply the lemma to $\hilb\oplus\hilb$, $P_N$ and $\widetilde{Q}=Q\oplus0$ instead of $\hilb$, $P$ and $Q$; in this case, subprojections of $\widetilde{Q}$ are of the form $F\oplus 0$, with $F$ a subprojection of $Q$. Since
%
\begin{equation*}
[P_N, \widetilde{Q}]=\left(\begin{array}{cc}[N,Q] & -Q\sqrt{NN'} \\ \sqrt{NN'}Q & 0\end{array}\right)
\end{equation*}
%
the hypothesis~(\ref{lmahyp}) of Lemma~\ref{lmahyp0} is fulfilled. The
claim yields
%
\begin{equation}
\label{traceconv}
\|[F_n-Q,N]\|_1\stackrel{n\to\infty}{\longrightarrow}0\,,
\end{equation}
%
as well as $\|\sqrt{NN'}(F_n-Q)\|_1\rightarrow0$, which however is already
known by~(\ref{mixedcond}) and $F_n=F_nQ$. We thus have a sequence $(F_n)$ of
unitarily implementable transformations: the conditions~(\ref{condcase1}) are
both fulfilled, the first one because $[N,\exp(-\iu\lambda
F_n)]=[N,F_n](\exp(-\iu\lambda)-1)$ and the second because
$(\Id-\exp(-\iu\lambda
F_n))\sqrt{NN'}=(\exp(-\iu\lambda)-1)F_nQ\sqrt{NN'}$. Moreover, the
assumptions of Prop.~\ref{continuity} are satisfied, so that
eqs.~(\ref{scalprodconv},\ref{approxscal}) are true again.

To complete the proof, it remains to show that
%
\begin{equation}
\label{detConvMix}
\det(N'+\ep{\iu\lambda F_{nU}}\ep{-\iu\lambda F_n}N)\longrightarrow\det(\ep{-\iu\lambda N_U Q_U}N'\ep{\iu\lambda N Q}+\ep{\iu\lambda N'_U Q_U}N\ep{-\iu\lambda N' Q})\,.
\end{equation}
%
To this end, we multiply the determinant by
%
\begin{equation}
\label{mixedcasemultdet}
\det(1+F_{nU}(\ep{-\iu\lambda N_U}-1))\,,\quad\det(1+(\ep{\iu\lambda N}-1)F_{n})\,,
\end{equation}
%
from the left, resp. from the right. These factors would be identical to those in~(\ref{purecasemultdet}) if $F_n$ and $N$ commuted, which is however no longer the case. Also, their product is not $1$, but rather equals
%
\begin{multline}\label{detcorr}
\det(1+F_{nU}(\ep{-\iu\lambda N_U}-1))\cdot\det(1+(\ep{\iu\lambda N}-1)F_{n})\\
\begin{aligned}
&=\det(1+(\ep{\iu\lambda N}-1)F_n)\cdot\det(1+F_n(\ep{-\iu\lambda N}-1))\\
&=\det(1-F_n+\ep{\iu\lambda N}F_n\ep{-\iu\lambda N})\,,
\end{aligned}
\end{multline}
%
where
%
\begin{equation*}
\ep{\iu\lambda N}F_n\ep{-\iu\lambda N}-F_n=\iu\int_0^{\lambda}\,\ep{\iu sN}[N,F_n]\ep{-\iu s N}ds\in\ideal{1}
\end{equation*}
%
and
%
\begin{equation}
\label{supp}
\|\ep{\iu\lambda N}F_n\ep{-\iu\lambda N}-F_n\|_1\stackrel{n\to\infty}{\longrightarrow} 0
\end{equation}
%
by $[N,Q]=0$ and~(\ref{traceconv}). Therefore, (\ref{detcorr}) converges to
$1$ and it suffices to prove~(\ref{detConvMix}) with the l.h.s. multiplied
by~(\ref{mixedcasemultdet}). The determinant becomes that of
%
\begin{equation}
\label{multipieddet}
(1+F_{nU}(\ep{-\iu\lambda N_U}-1))(N'+\ep{\iu\lambda F_{nU}}\ep{-\iu\lambda F_n}N)(1+(\ep{\iu\lambda N}-1)F_n)\,.
\end{equation}
%
By means of
%
\begin{align}
\label{firstconv}
&\|(1+F_{nU}(\ep{-\iu\lambda N_U}-1))N'-\ep{-\iu\lambda N_UQ_U}N'\|_1\longrightarrow 0\,, \\
\label{secconv}
&\|\ep{-\iu\lambda F_n}N\ep{\iu\lambda F_n}-N\|_1\longrightarrow 0\,,
\end{align}
%
which we shall prove momentarily, we may replace~(\ref{multipieddet}) by
%
\begin{equation*}
\ep{-\iu\lambda N_UQ_U}N'(1+(\ep{\iu\lambda N}-1)F_n)+(1+F_{nU}(\ep{-\iu\lambda N_U}-1))\ep{\iu\lambda F_{nU}}N\ep{-\iu\lambda F_{n}}(1+(\ep{\iu\lambda N}-1)F_n)\,.
\end{equation*}
%
The claim then follows from
%
\begin{align}
\label{thirdconv}
&\|N'(1+(\ep{\iu\lambda N}-1)F_n)-N'\ep{\iu\lambda NQ}\|_1\longrightarrow 0\,, \\
\label{fourthconv}
&\|N\ep{-\iu\lambda F_n}(1+(\ep{\iu\lambda N}-1)F_n)-N\ep{-\iu\lambda N'Q}\|_1\longrightarrow 0\,, \\
\label{fifthconv}
&\|(1+F_{nU}(\ep{-\iu\lambda N_U}-1))\ep{\iu\lambda F_{nU}}N-\ep{\iu\lambda N'_UQ_U}N\|_1\longrightarrow 0\,,
\end{align}
%
It remains to prove (\ref{firstconv}\,-\,\ref{fifthconv}). The limit~(\ref{secconv}) follows like~(\ref{supp}). The expression in~(\ref{thirdconv}) is $N'(\exp(\iu\lambda N)-1)(F_n-Q)=f(N)\,NN'Q(F_n-Q)$ where $f(N)=N^{-1}(\exp(\iu\lambda N)-1)$ is a bounded operator; its convergence to zero follows from~(\ref{mixedcond}). As for~(\ref{fourthconv}) we have
%
\begin{align*}
\ep{-\iu\lambda F_n}(1+(\ep{\iu\lambda N}-1)F_n)&=(1+(\ep{-\iu\lambda}-1)F_n)(1+(\ep{\iu\lambda N}-1)F_n) \\
&=1+(\ep{-\iu\lambda}\ep{\iu\lambda N}-1)F_n+(\ep{-\iu\lambda}-1)[F_n,\ep{\iu\lambda N}] F_n\,,
\end{align*}
%
so that by using~(\ref{supp}) it remains to show
\begin{equation*}
\|N(1+(\ep{-\iu\lambda N'}-1)F_n)-N\ep{-\iu\lambda N'Q}\|_1\longrightarrow 0\,.
\end{equation*}
%
This, however, is just~(\ref{thirdconv}) with $N$ and $N'$ interchanged. Finally, in~(\ref{firstconv}, \ref{fifthconv}) we may, by~(\ref{NTrCl}), replace $N$ and $N'$ by $N_U$ and $N_U'$ in those places where the subscript is not already present. By passing to a unitary conjugate and adjoint, they reduce to~(\ref{thirdconv}, \ref{fourthconv}).\teop

\section{Examples}\label{exa}
We illustrate the hypotheses~(\ref{QN}, \ref{NTrCl}) by presenting a model in
which they can be verified. The left and right portions of the single lead
mentioned in Sect.~1 are replaced by two infinite leads, which are however
chiral. The interaction between them occurs in a finite interval and allows
particles to scatter between the leads.

\begin{figure}[h]
\begin{center}
\input{Pump_ex.pstex_t}
\caption{A simple model with two infinite chiral leads}\label{fig_chiralPump}
\end{center}
\end{figure}

Let $\hilb=L^2(\mathbb{R})\oplus L^2(\mathbb{R})$ be the one-particle space with operators
%
\begin{equation}
Q=\left(\begin{array}{cc}
1 & 0 \\ 0 & 0
\end{array}\right)\,,\qquad N=\left(\begin{array}{cc}
\Theta(-p) & 0 \\ 0 & \Theta(p)
\end{array}\right)\,.
\end{equation}
%
Here, $x\in\mathbb{R}$ is the position variable, $p=-\iu d/dx$ the conjugate
momentum, and $\Theta$ the Heaviside function. The projection $N$ describes
the Fermi sea of the free Hamiltonian
%
\begin{equation}
\label{ex1_hamiltonian}
H_0=\left(\begin{array}{cc}
p & 0 \\ 0 & -p
\end{array}\right)
\end{equation}
%
for vanishing Fermi energy. Clearly, $[Q,N]=0$.

Our first example is conveniently stated by passing to another pair of
conjugate variables, $E$ and $t$: The energy $E=\pm p$ yields the spectral
representation of $H_0$ in multiplication form,
%
\[
H_0=\left(\begin{array}{cc}
E & 0 \\ 0 & E
\end{array}\right)\,,
\]
%
while the operator
%
\[
T=\left(\begin{array}{cc}
-x & 0 \\ 0 & x
\end{array}\right)
\equiv
\left(\begin{array}{cc}
t & 0 \\ 0 & t
\end{array}\right)\,,
\]
%
represents, due to $\iu [H_0,T]=-1$, the time $t$ of passage at $x=0$ of
a freely moving particle.

\subsection{Example 1}\label{Example 1}

Rather than specifying an interacting Hamiltonian, we model the
scattering process by directly giving the propagator $U$ for the
time interval under consideration. We assume it to be given by a
unitary multiplication operator $U(t)$ with $U(t)-1$ of compact
support, see Remark~1 in Sect.~2.

Such a simple kind of evolution should be seen as an effective description in
the adiabatic limit. The passage across the
interaction region maps the incoming state to the outgoing one
by means of a scattering matrix which, in the limit of low frequencies
$\omega$, is that of the static scatterer in effect at time $t$, $S(t)$. In
the same limit, only electrons within an interval $\sim\hbar\omega$ of the
Fermi energy ought to matter for the transport. Thus, $U(t)=S(t,0)$, where
the $2\times 2$-matrix $S(t,E)$ is the fiber of $S(t)$ at energy $E$. For a
more thorough justification, see~\cite{BPT, AEGS}.
%
\begin{prop}
\label{exa1}
Suppose $U-1\in M_2(C_0^{\infty}(\mathbb{R}_t))$. Then $[N,U]\in \ideal{1}$.
\end{prop}
%
Here $M_2(X)$ are the $2\times 2$ matrices with entries in $X$.

\paragraph{Proof.} We may rename $U(t)-1$ by $U(t)$
without loss. By the assumption we may write
$U=fU$, where $f=f(t)$ satisfies $f\in
C_0^{\infty}(\mathbb{R})$, too. Then
$[N,U]=f[N,U]+[N,f]U$, with
\begin{equation*}
f[N,U]=f(E+\iu)^{-1}\cdot(E+\iu)[N,U]\,,
\end{equation*}
and similarly for the second term. Both factors are Hilbert-Schmidt and hence
their product trace class. The first one is, because the functions $f$ and
$g(E)\equiv (E+\iu)^{-1}$ are in $L^2(\mathbb{R})$. As for the second
one, we note that $N=\Theta(-E)\otimes 1_2$, whence $[N,U]$ has
matrix entries $[\Theta(-E),U_{ij}]$. That leads to integral
operators $K$ acting merely on $L^2(\mathbb{R})$ with kernels
\begin{equation*}
K(E,E')=(E+\iu)\widehat{U}_{ij}(E'-E)\bigl(\Theta(-E)-\Theta(-E')\bigr)\,.
\end{equation*}
They are supported where $\mathrm{sgn}\,E = -\mathrm{sgn}\,E'$ and satisfy
\begin{equation*}
\iint |K(E,E')|^2 dE\,dE'=
\int_0^{\infty}\int_0^{\infty}|E''+\iu|^2\bigl(
|\widehat{U}_{ij}(E'+E'')|^2 +|\widehat{U}_{ij}(-E'-E'')|^2\bigr)dE\,dE''\,,
\end{equation*}
which is finite. Thus, the corresponding operator is in
$\ideal{2}$.\teop\newline

By contrast, but under the same assumption as in the proposition, the
operator $(Q_U-Q)N$ may fail to be trace class. By~(\ref{ExpTransp}), this
shows the need for regularizing~(\ref{Levitov}). Indeed, we may arrange for a
$\psi\in\hilb$ and $U$ such that $(Q_U-Q)\psi\neq 0$. The sequence
$\psi_n=\exp(\iu n t)\psi$ tends to zero weakly. Using
$\Theta(-E)\exp(\iu n t)=\exp(\iu n t)\Theta(n-E)$ and
$\Theta(n-E)\stackrel{s}{\rightarrow}1$, we
have $\|N\psi_n-\psi_n\|\rightarrow 0$ and, since $Q$, $U$ are multiplication
operators in $t$, $\|(Q_U-Q)N\psi_n\|\rightarrow\|(Q_U-Q)\psi\|\neq 0$. As a
result, $(Q_U-Q)N$ is not even compact.
The argument just given may be summarized in physical terms as follows: 
Whatever contribution to transport, as signified by $(Q_U-Q)N$, comes from one
energy in the Fermi sea, it is repeated at all such energies, because the
evolution $U$ proceeds with the same velocity $\pm 1$ at all energies. 

It should be remarked that $[N,U]$ may fail to be in
$\ideal{2}$ if, unlike in Prop.~\ref{exa1}, $U(t)$ attains different
limits at $t\to\pm\infty$. This fact has been pointed out in
\cite{LevitovLeeLesovik} in slightly different terms as a
manifestation of the orthogonality catastrophe. Consider for
instance a potential drop $V(t)$ of finite duration being applied
between the leads, with $\int_{-\infty}^\infty V(t)dt\notin
2\pi\mathbb{Z}$. That situation can be modeled in the context of the
present example by means of a vector potential, where it gives raise
to the catastrophe. The same physical situation is however tame in
the context of the next example.

\subsection{Example 2}

Here we specify a time-dependent perturbation of (\ref{ex1_hamiltonian}),
$H(t)=H_0+V(t)$, where $V(t)$ is multiplication by a $2\times 2$ matrix
$V(t,x)$. Let $U=U(t_2,t_1)$ be the propagator for $H(t)$ between times $t_1$
and $t_2$.
%
\begin{prop}
Suppose
$V(t,\cdot),\,\partial_t V(t,\cdot)\in M_2(C_0^{\infty}(\mathbb{R}_x))$.
Then $[N,U]\in \ideal{2}$.
\end{prop}
%
Note that the commutator is claimed to be Hilbert-Schmidt only, which covers
only the statements (ii-iv) of Theorem~\ref{thm1}.
%
\paragraph{Proof.} By \cite{RSe}, Lemma 4  or \cite{Ru}, Thm. 2.8 it suffices
to show that the statement holds true for the first term in the Dyson
expansion of $U$, \ie for
%
\begin{equation}
\label{dyson}
\tilde U(s_2,s_1)=-\iu\int_{s_1}^{s_2}\ep{\iu H_0t}V(t)\ep{-\iu H_0t} dt \,,
\end{equation}
%
with estimates uniform in the sub-interval $[s_1,s_2]\subset[t_1,t_2]$.
By writing
%
\[
V(t)=\left(\begin{array}{cc}
V_{++}(t)&V_{+-}(t)\\
V_{-+}(t)&V_{--}(t)
\end{array}\right)\,,
\]
%
the kernel of $[N,V(t)]$ in momentum space becomes
%
\[
[N,V(t)](p,p')=
\left(\begin{array}{cc}
\hat V_{++}(t,p-p')(\Theta(-p)-\Theta(-p'))&
\hat V_{+-}(t,p-p')(\Theta(-p)-\Theta(p'))\\
\hat V_{-+}(t,p-p')(\Theta(p)-\Theta(-p'))&
\hat V_{--}(t,p-p')(\Theta(p)-\Theta(p'))
\end{array}\right)\,.
\]
%
The diagonal contributions are in $\ideal{2}$ without recourse to the
integration (\ref{dyson}). For instance,
%
\[
\iint dpdp'\,|\hat V_{--}(t,p-p')|^2|\Theta(p)-\Theta(p')|
%=\Bigl(\int_0^\infty\int_{-\infty}^0+\int_{-\infty}^0\int_0^\infty\Bigr)
%dpdp'\, |\hat V_{--}(t,p-p')|^2
=\int_{-\infty}^\infty du\,|u||\hat V_{--}(t,u)|^2<\infty\,.
\]
%
The off-diagonal contributions improve once the time integral is performed.
We compute it by parts and obtain, for instance, the kernel
%
\begin{multline}
\label{pint}
-\iu\int_{s_1}^{s_2}\hat V_{+-}(t,p-p')
\ep{\iu(p+p')t}dt\\
=-\frac{\ep{\iu(p+p')t}-1}{p+p'}\hat V_{+-}(t,p-p')\Big|_{s_1}^{s_2}
+\int_{s_1}^{s_2}\frac{\ep{\iu(p+p')t}-1}{p+p'}
\partial_t \hat V_{+-}(t,p-p')dt\,,
\end{multline}
%
times $\Theta(-p)-\Theta(p')$. The boundary terms are separately in
$\ideal{2}$, since their corresponding square norm is
 %
\begin{multline*}
4\iint dpdp'\,\frac{\sin^2((p+p')s_i/2)}{(p+p')^2}
|\hat V_{+-}(s_i,p-p')|^2|\Theta(-p)-\Theta(p')|\\
%=4\Bigl(\int_0^\infty\int_0^\infty+\int_{-\infty}^0\int_{-\infty}^0\Bigr)
%dpdp'\,|\frac{\sin^2((p+p')s_i/2)}{(p+p')^2}
%|\hat V_{+-}(s_i,p-p')|^2\\
=4\int_{-\infty}^\infty du\,\frac{\sin^2(us_i/2)}{u^2}
\int_{-|u|/2}^{|u|/2}dv\, |\hat V_{+-}(s_i,v)|^2
\le \pi |s_i| \|V_{+-}(s_i)\|_2^2\,.
\end{multline*}
%
By the same estimate, but with $\partial_t V_{+-}(t)$ in place of
$V_{+-}(s_i)$, also the integrand in (\ref{pint}) is in
$\ideal{2}$. \teop\newline

We recall that in \cite{RSe, Ru} the implementation of the propagator of a
time-dependent Dirac Hamiltonian was studied, of which the above $H(t)$ is the
1-dimensional version. In larger dimensions, as considered there, the
implementability is ensured only in some cases.

We remark that by the method used in Example 1 one can show that
diagonal perturbations lead to $[N,U]\in \ideal{1}$, but not for off-diagonal
ones.



\paragraph{Acknowledgements.} We would like to thank H. Araki,
P. Deift, G. Dell'Antonio, G. Kottanattu and G. Lesovik for discussions.
We also thank the Erwin Schr\"odinger Institute (Vienna) and the
Lewiner Institute for Theoretical Physics at the Technion (Haifa) for
hospitality.

%\bibliography{PumpBib8}
%\bibliographystyle{plain}

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  0 1 4 index length 2 sub {
				% string proc char idx
    % Find the n+1th character in the string
    3 index exch 1 add get	% string proc char char+1
    exch 2 copy			% strinq proc char+1 char char+1 char
    % Now show the nth character
    PATsstr dup 0 4 -1 roll put	% string proc chr+1 chr chr+1 (chr)
    false charpath		% string proc char+1 char char+1
    /clip load PATdraw
    % Move past the character (charpath modified the current point)
    currentpoint newpath moveto
    % Execute the user proc (should consume char and char+1)
    mark 3 1 roll		% string proc char+1 mark char char+1
    4 index exec		% string proc char+1 mark...
    cleartomark			% string proc char+1
  } for
  % Now display the last character
  PATsstr dup 0 4 -1 roll put	% string proc (char+1)
  false charpath		% string proc
  /clip load PATdraw
  neewath
  pop pop			% -
} bind def
% PATmp - the makepattern equivalent
/PATmp {			% patdict patmtx PATmp patinstance
  exch dup length 7 add		% We will add 6 new entries plus 1 FID
  dict copy			% Create a new dictionary
  begin
    % Matrix to install when painting the pattern
    TilingType PATtcalc
    /PatternGState PATcg def
    PatternGState /cm 3 -1 roll put
    % Check for multi pattern sources (Level 1 fast color patterns)
    currentdict /Multi known not { /Multi 1 def } if
    % Font dictionary definitions
    /FontType 3 def
    % Create a dummy encoding vector
    /Encoding 256 array def
    3 string 0 1 255 {
      Encoding exch dup 3 index cvs cvn put } for pop
    /FontMatrix matrix def
    /FontBBox BBox def
    /BuildChar {
	mark 3 1 roll		% mark dict char
	exch begin
	Multi 1 ne {PaintData exch get}{pop} ifelse  % mark [paintdata]
	  PaintType 2 eq Multi 1 ne or
	  { XStep 0 FontBBox aload pop setcachedevice }
	  { XStep 0 setcharwidth } ifelse
	  currentdict		% mark [paintdata] dict
	  /PaintProc load	% mark [paintdata] dict paintproc
	end
	gsave
	  false PATredef exec true PATredef
	grestore
	cleartomark		% -
    } bind def
    currentdict
  end				% newdict
  /foo exch			% /foo newlict
  definefont			% newfont
} bind def
% PATpcalc - calculates the starting point and width/height
% of the tile fill for the shape
/PATpcalc {	% - PATpcalc nw nh px py
  PATDict /CurrentPattern get begin
    gsave
	% Set up the coordinate system to Pattern Space
	% and lock down pattern
	PatternGState /cm get setmatrix
	BBox aload pop pop pop translate
	% Determine the bounding box of the shape
	pathbbox			% llx lly urx ury
    grestore
    % Determine (nw, nh) the # of cells to paint width and height
    PatHeight div ceiling		% llx lly urx qh
    4 1 roll				% qh llx lly urx
    PatWidth div ceiling		% qh llx lly qw
    4 1 roll				% qw qh llx lly
    PatHeight div floor			% qw qh llx ph
    4 1 roll				% ph qw qh llx
    PatWidth div floor			% ph qw qh pw
    4 1 roll				% pw ph qw qh
    2 index sub cvi abs			% pw ph qs qh-ph
    exch 3 index sub cvi abs exch	% pw ph nw=qw-pw nh=qh-ph
    % Determine the starting point of the pattern fill
    %(px, py)
    4 2 roll				% nw nh pw ph
    PatHeight mul			% nw nh pw py
    exch				% nw nh py pw
    PatWidth mul exch			% nw nh px py
  end
} bind def

% Save the original routines so that we can use them later on
/oldfill	/fill load def
/oldeofill	/eofill load def
/oldstroke	/stroke load def
/oldshow	/show load def
/oldashow	/ashow load def
/oldwidthshow	/widthshow load def
/oldawidthshow	/awidthshow load def
/oldkshow	/kshow load def

% These defs are necessary so that subsequent procs don't bind in
% the originals
/fill	   { oldfill } bind def
/eofill	   { oldeofill } bind def
/stroke	   { oldstroke } bind def
/show	   { oldshow } bind def
/ashow	   { oldashow } bind def
/widthshow { oldwidthshow } bind def
/awidthshow { oldawidthshow } bind def
/kshow 	   { oldkshow } bind def
/PATredef {
  MyAppDict begin
    {
    /fill { /clip load PATdraw newpath } bind def
    /eofill { /eoclip load PATdraw newpath } bind def
    /stroke { PATstroke } bind def
    /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def
    /ashow { 0 0 null 6 3 roll PATawidthshow }
    bind def
    /widthshow { 0 0 3 -1 roll PATawidthshow }
    bind def
    /awidthshow { PATawidthshow } bind def
    /kshow { PATkshow } bind def
  } {
    /fill   { oldfill } bind def
    /eofill { oldeofill } bind def
    /stroke { oldstroke } bind def
    /show   { oldshow } bind def
    /ashow  { oldashow } bind def
    /widthshow { oldwidthshow } bind def
    /awidthshow { oldawidthshow } bind def
    /kshow  { oldkshow } bind def
    } ifelse
  end
} bind def
false PATredef
% Conditionally define setcmykcolor if not available
/setcmykcolor where { pop } {
  /setcmykcolor {
    1 sub 4 1 roll
    3 {
	3 index add neg dup 0 lt { pop 0 } if 3 1 roll
    } repeat
    setrgbcolor - pop
  } bind def
} ifelse
/PATsc {		% colorarray
  aload length		% c1 ... cn length
    dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor
  } ifelse } ifelse
} bind def
/PATsg {		% dict
  begin
    lw setlinewidth
    lc setlinecap
    lj setlinejoin
    ml setmiterlimit
    ds aload pop setdash
    cc aload pop setrgbcolor
    cm setmatrix
  end
} bind def

/PATDict 3 dict def
/PATsp {
  true PATredef
  PATDict begin
    /CurrentPattern exch def
    % If it's an uncolored pattern, save the color
    CurrentPattern /PaintType get 2 eq {
      /PColor exch def
    } if
    /CColor [ currentrgbcolor ] def
  end
} bind def
% PATstroke - stroke with the current pattern
/PATstroke {
  countdictstack
  save
  mark
  {
    currentpoint strokepath moveto
    PATpcalc				% proc nw nh px py
    clip newpath PATfill
    } stopped {
	(*** PATstroke Warning: Path is too complex, stroking
	  with gray) =
    cleartomark
    restore
    countdictstack exch sub dup 0 gt
	{ { end } repeat } { pop } ifelse
    gsave 0.5 setgray oldstroke grestore
  } { pop restore pop } ifelse
  newpath
} bind def
/PATtcalc {		% modmtx tilingtype PATtcalc tilematrix
  % Note: tiling types 2 and 3 are not supported
  gsave
    exch concat					% tilingtype
    matrix currentmatrix exch			% cmtx tilingtype
    % Tiling type 1 and 3: constant spacing
    2 ne {
	% Distort the pattern so that it occupies
	% an integral number of device pixels
	dup 4 get exch dup 5 get exch		% tx ty cmtx
	XStep 0 dtransform
	round exch round exch			% tx ty cmtx dx.x dx.y
	XStep div exch XStep div exch		% tx ty cmtx a b
	0 YStep dtransform
	round exch round exch			% tx ty cmtx a b dy.x dy.y
	YStep div exch YStep div exch		% tx ty cmtx a b c d
	7 -3 roll astore			% { a b c d tx ty }
    } if
  grestore
} bind def
/PATusp {
  false PATredef
  PATDict begin
    CColor PATsc
  end
} bind def

% right30
11 dict begin
/PaintType 1 def
/PatternType 1 def
/TilingType 1 def
/BBox [0 0 1 1] def
/XStep 1 def
/YStep 1 def
/PatWidth 1 def
/PatHeight 1 def
/Multi 2 def
/PaintData [
  { clippath } bind
  { 32 16 true [ 32 0 0 -16 0 16 ]
	{<00030003000c000c0030003000c000c0030003000c000c00
	30003000c000c00000030003000c000c0030003000c000c0
	030003000c000c0030003000c000c000>}
     imagemask } bind
] def
/PaintProc {
	pop
	exec fill
} def
currentdict
end
/P2 exch def

% horizontal lines
11 dict begin
/PaintType 1 def
/PatternType 1 def
/TilingType 1 def
/BBox [0 0 1 1] def
/XStep 1 def
/YStep 1 def
/PatWidth 1 def
/PatHeight 1 def
/Multi 2 def
/PaintData [
  { clippath } bind
  { 16 8 true [ 16 0 0 -8 0 8 ]
	{< ffff000000000000ffff000000000000>}
     imagemask } bind
] def
/PaintProc {
	pop
	exec fill
} def
currentdict
end
/P9 exch def

% vertical lines
11 dict begin
/PaintType 1 def
/PatternType 1 def
/TilingType 1 def
/BBox [0 0 1 1] def
/XStep 1 def
/YStep 1 def
/PatWidth 1 def
/PatHeight 1 def
/Multi 2 def
/PaintData [
  { clippath } bind
  { 8 16 true [ 8 0 0 -16 0 16 ]
	{<11111111111111111111111111111111>}
     imagemask } bind
] def
/PaintProc {
	pop
	exec fill
} def
currentdict
end
/P10 exch def

/cp {closepath} bind def
/ef {eofill} bind def
/gr {grestore} bind def
/gs {gsave} bind def
/sa {save} bind def
/rs {restore} bind def
/l {lineto} bind def
/m {moveto} bind def
/rm {rmoveto} bind def
/n {newpath} bind def
/s {stroke} bind def
/sh {show} bind def
/slc {setlinecap} bind def
/slj {setlinejoin} bind def
/slw {setlinewidth} bind def
/srgb {setrgbcolor} bind def
/rot {rotate} bind def
/sc {scale} bind def
/sd {setdash} bind def
/ff {findfont} bind def
/sf {setfont} bind def
/scf {scalefont} bind def
/sw {stringwidth} bind def
/tr {translate} bind def
/tnt {dup dup currentrgbcolor
  4 -2 roll dup 1 exch sub 3 -1 roll mul add
  4 -2 roll dup 1 exch sub 3 -1 roll mul add
  4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb}
  bind def
/shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul
  4 -2 roll mul srgb} bind def
/$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def
/$F2psEnd {$F2psEnteredState restore end} def

$F2psBegin
10 setmiterlimit
0 slj 0 slc
 0.06299 0.06299 sc
%
% Fig objects follow
%
% 
% here starts figure with depth 53
% Polyline
0 slj
0 slc
0.000 slw
n 2700 540 m 5850 540 l 5850 720 l 2700 720 l
 cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def
15.00 15.00 sc P2 [16 0 0 -8 180.00 36.00] PATmp PATsp ef gr PATusp 
% Polyline
n 2700 1980 m 5850 1980 l 5850 2160 l 2700 2160 l
 cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def
15.00 15.00 sc P2 [16 0 0 -8 180.00 132.00] PATmp PATsp ef gr PATusp 
% Polyline
n 5310 2520 m 3600 180 l 3420 180 l 5130 2520 l
 cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def
15.00 15.00 sc P10 [8 0 0 -16 228.00 12.00] PATmp PATsp ef gr PATusp 
% Polyline
n 4275 180 m 4500 180 l 4500 2520 l 4275 2520 l
 cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def
15.00 15.00 sc P9 [16 0 0 -8 285.00 12.00] PATmp PATsp ef gr PATusp 
% Polyline
7.500 slw
gs  clippath
930 603 m 930 435 l 870 435 l 870 603 l 870 603 l 900 483 l 930 603 l cp
eoclip
n 900 2250 m
 900 450 l gs col0 s gr gr

% arrowhead
n 930 603 m 900 483 l 870 603 l  col0 s
% Polyline
gs  clippath
1647 750 m 1815 750 l 1815 690 l 1647 690 l 1647 690 l 1767 720 l 1647 750 l cp
eoclip
n 0 720 m
 1800 720 l gs col0 s gr gr

% arrowhead
n 1647 750 m 1767 720 l 1647 690 l  col0 s
% Polyline
gs  clippath
5697 1380 m 5865 1380 l 5865 1320 l 5697 1320 l 5697 1320 l 5817 1350 l 5697 1380 l cp
eoclip
n 2700 1350 m
 5850 1350 l gs col0 s gr gr

% arrowhead
n 5697 1380 m 5817 1350 l 5697 1320 l  col0 s
% Polyline
gs  clippath
4305 333 m 4305 165 l 4245 165 l 4245 333 l 4245 333 l 4275 213 l 4305 333 l cp
eoclip
n 4275 2520 m
 4275 180 l gs col0 s gr gr

% arrowhead
n 4305 333 m 4275 213 l 4245 333 l  col0 s
% Polyline
n 2700 540 m
 5850 540 l gs col0 s gr 
% Polyline
n 2700 2160 m
 5850 2160 l gs col0 s gr 
% Polyline
n 5130 2520 m
 3420 180 l gs col0 s gr 
% Polyline
 [60] 0 sd
n 1700 180 m
 1335 2070 l gs col0 s gr  [] 0 sd
% Polyline
 [60] 0 sd
n 100 180 m
 465 2070 l gs col0 s gr  [] 0 sd
% Polyline
2 slj
n 180 540 m 180 543 l 181 549 l 183 560 l 186 577 l 189 600 l
 194 629 l 199 662 l 205 699 l 211 737 l 218 777 l
 224 817 l 231 856 l 238 893 l 244 929 l 250 963 l
 256 995 l 262 1025 l 268 1053 l 274 1081 l 281 1107 l
 287 1133 l 293 1159 l 300 1185 l 307 1211 l 314 1237 l
 322 1264 l 330 1291 l 339 1318 l 347 1346 l 357 1375 l
 366 1404 l 376 1433 l 386 1462 l 397 1491 l 407 1520 l
 418 1548 l 428 1576 l 439 1603 l 449 1629 l 460 1654 l
 470 1679 l 479 1702 l 489 1723 l 498 1744 l 507 1764 l
 516 1782 l 525 1800 l 538 1825 l 550 1848 l 563 1871 l
 576 1892 l 589 1912 l 602 1930 l 615 1948 l 628 1964 l
 641 1979 l 653 1993 l 665 2005 l 677 2016 l 688 2025 l
 699 2033 l 710 2041 l 720 2048 l 734 2056 l 748 2063 l
 763 2070 l 778 2077 l 794 2082 l 810 2087 l 826 2091 l
 842 2095 l 857 2097 l 872 2099 l 886 2100 l 900 2100 l
 914 2100 l 928 2099 l 943 2097 l 958 2095 l 974 2091 l
 990 2087 l 1006 2082 l 1022 2077 l 1037 2070 l 1052 2063 l
 1066 2056 l 1080 2048 l 1090 2041 l 1101 2033 l 1112 2025 l
 1123 2016 l 1135 2005 l 1147 1993 l 1159 1979 l 1172 1964 l
 1185 1948 l 1198 1930 l 1211 1912 l 1224 1892 l 1237 1871 l
 1250 1848 l 1262 1825 l 1275 1800 l 1284 1782 l 1293 1764 l
 1302 1744 l 1311 1723 l 1321 1702 l 1330 1679 l 1340 1654 l
 1351 1629 l 1361 1603 l 1372 1576 l 1382 1548 l 1393 1520 l
 1403 1491 l 1414 1462 l 1424 1433 l 1434 1404 l 1443 1375 l
 1453 1346 l 1461 1318 l 1470 1291 l 1478 1264 l 1486 1237 l
 1493 1211 l 1500 1185 l 1507 1159 l 1513 1133 l 1519 1107 l
 1526 1081 l 1532 1053 l 1538 1025 l 1544 995 l 1550 963 l
 1556 929 l 1562 893 l 1569 856 l 1576 817 l 1582 777 l
 1589 737 l 1595 699 l 1601 662 l 1606 629 l 1611 600 l
 1614 577 l 1617 560 l 1619 549 l 1620 543 l
 1620 540 l gs col0 s gr 
% here ends figure;
$F2psEnd
rs
end
showpage
%%Trailer
%EOF
---------------0705010715578--
