Content-Type: multipart/mixed; boundary="-------------0705010715578" This is a multi-part message in MIME format. ---------------0705010715578 Content-Type: text/plain; name="07-109.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="07-109.keywords" counting statistics, Fredholm determinants ---------------0705010715578 Content-Type: application/x-tex; name="Pump_ex.pstex_t" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Pump_ex.pstex_t" \begin{picture}(0,0)% \includegraphics{Pump_ex.pstex}% \end{picture}% \setlength{\unitlength}{3108sp}% % \begingroup\makeatletter\ifx\SetFigFont\undefined% \gdef\SetFigFont#1#2#3#4#5{% \reset@font\fontsize{#1}{#2pt}% \fontfamily{#3}\fontseries{#4}\fontshape{#5}% \selectfont}% \fi\endgroup% \begin{picture}(4524,1220)(3049,-3326) \end{picture}% ---------------0705010715578 Content-Type: application/x-tex; name="counting8.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="counting8.tex" \documentclass[12pt]{article} \usepackage{amsmath} \usepackage{amsthm} \usepackage{amssymb} \usepackage{mathrsfs} %\usepackage{psfrag} \usepackage{bbm} %\usepackage{epic} \usepackage{graphicx,color} %\usepackage[pdftex]{graphicx, color} \renewcommand{\baselinestretch}{1.1} %\numberwithin{equation}{section} \addtolength{\textwidth}{3cm} \addtolength{\hoffset}{-1.5cm} %\include{mymacros} \newcommand{\cf}{{\it cf.\/} } \newcommand{\etal}{{\it et.\/ al.\/} } \newcommand{\ie}{{\it i.e.\/} } \newcommand{\eg}{{\it e.g.\/} } \newcommand{\Ran}[1]{\mathrm{Ran}\, #1 } \newcommand{\Ker}[1]{\mathrm{Ker}\, #1 } \newcommand{\eps}{\epsilon} \newcommand{\iu}{\mathrm{i}} \newcommand{\Id}{1} \newcommand{\alg}[1]{\mathcal{A}(#1)} \newcommand{\ideal}[1]{\mathcal{I}_{ #1 }} \newcommand{\str}{^{*}} \newcommand{\ep}[1]{\mathrm{e}^{#1}} \newcommand{\hilb}{\mathcal{H}} \newcommand{\dd}{\mathrm{d}} \newcommand{\tr}{\mathrm{tr}} \newcommand{\Tr}{\mathrm{Tr}} \newcommand{\meop}{\qquad\blacklozenge} \newcommand{\teop}{\hfill$\square$} \newtheorem{rem}{Remark} \newtheorem{thm}{Theorem} \newtheorem{cor}[thm]{Corollary} \newtheorem{prop}[thm]{Proposition} \newtheorem{lma}[thm]{Lemma} \newtheorem{defin}{Definition} \newcommand{\En}{\mathcal{E}} \renewcommand{\labelenumi}{\roman{enumi}\/.} %\setcounter{figure}{0} %\newcommand{\red}{\color{red}} %\newcommand{\blk}{\color{black}} %\newcommand{\blu}{\color{blue}} %\newcommand{\grn}{\color{green}} \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 $|p|0$. 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 $0\mathrm{rank}(U-1)$ 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 $i_1<\ldots0$. 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. 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Mech.}, 14:315--322, 1965. \end{thebibliography} \end{document} ---------------0705010715578 Content-Type: application/x-tex; name="figure1.pstex_t" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="figure1.pstex_t" \begin{picture}(0,0)% \includegraphics{figure1.pstex}% \end{picture}% \setlength{\unitlength}{4144sp}% % \begingroup\makeatletter\ifx\SetFigFont\undefined% \gdef\SetFigFont#1#2#3#4#5{% \reset@font\fontsize{#1}{#2pt}% \fontfamily{#3}\fontseries{#4}\fontshape{#5}% \selectfont}% \fi\endgroup% \begin{picture}(5874,2421)(-11,-1714) \put(4051,524){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$p$}% }}}} \put(5761,-691){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$x$}% }}}} \put(2791,-61){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$N$}% }}}} 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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 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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 ] 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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--