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\begin{document}
\title{\textbf{New product formul{\ae}
and quantum Zeno dynamics with generalized observables}}
\author{Pavel Exner$^{a}$, Takashi Ichinose$^{b}$, Hagen Neidhardt$^{c}$, \\and\\
Valentin A. Zagrebnov$^{d}$}
\date{\today}
\maketitle \maketitle
\begin{quote}
{\small \em a) Department of Theoretical Physics, NPI, Academy of
Sciences, \mbox{CZ-25068} \v Re\v z, and Doppler Institute, Czech
Technical University, B\v{r}ehov{\'a}~7,
CZ-11519 Prague, Czech Republic \\
b) Department of Mathematics, Faculty of Science, Kanazawa
University, Kanazawa 920-1192,
Japan\\
c) Weierstra{\ss}-Institut f\"ur Angewandte Analysis und
Stochastik,
Mohrenstr. 39, D-10117 Berlin, Germany \\
d) D\'{e}partment de Physique, Universit\'{e} de la
M\'{e}diterran\'{e}e (Aix-Marseille II) and Centre de Physique
Th\'{e}orique, CNRS, Luminy Case 907, F-13288
Marseille Cedex 9, France \\
\rm exner@ujf.cas.cz, ichinose@kappa.s.kanazawa-u.ac.jp, neidhard@wias-berlin.de,
zagrebnov@cpt.univ-mrs.fr} \vspace{8mm}
%\noindent \textbf{Abstract:}
\end{quote}
\begin{abstract}
We demonstrate a pair of new product formul{\ae} which combine a
projection with a resolvent of a positive operator, or with an
exponential function and spectral projections, respectively. The
convergence is strong or even operator-norm under more restrictive
assumptions. The second mentioned formula can be used to describe
Zeno dynamics in the situation when the usual non-decay
measurement is replaced by a particular generalized observable in
the sense of Davies.
\end{abstract}
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
Product formul{\ae} are a traditional tool in various branches of
mathematics; their use dates back to the time of Sophus Lie. In
more recent times most attention was paid to Trotter formula
combining two semigroups or unitary groups, and its various
generalizations, motivated in part by the usefulness of this tool
in functional integration, quantum statistical physics and other
parts of physics -- see, e.g., \cite[Chap.~V]{Ex}, \cite{Za} and
references therein. The last decade brought a progress, with the
participation of some of the present authors, in understanding of
the convergence properties of such formul{\ae} -- for a review of
these results we refer to the monograph \cite{Za}.
In the last few years we have witnessed a surge of interest to
another type of product formul{\ae} in which a (semi)group is
combined with a projection operator motivated by the ``quantum
Zeno effect'' (QZE). This is also a venerable problem known
already to Alan Turing and formulated in the usual decay context
for the first time by Beskow and Nilsson \cite{BN}: frequent
measurements can slow down a decay of an unstable system, or even
fully stop it in the limit of infinite measurement frequency. The
effect was analyzed mathematically by Friedman \cite{Fr} but
became popular only after the authors of \cite{MS} invented the
above stated name. Recent interest is motivated mainly by the fact
that now the effect is within experimental reach; an up-to-date
bibliography can be found, e.g., in \cite{FMP} or \cite{Sch}.
At the same time there are still important unanswered questions.
The central among them concerns the existence and properties of
the Zeno dynamics. To explain it, recall the usual way in which
quantum kinematics of decays is described \cite[Chap.~I]{Ex}. The
unstable system is characterized by a projection $P$ to a subspace
in the state Hilbert space $\gotH$ of a larger, isolated system,
the dynamics of which is governed by a self-adjoint Hamiltonian
$H$. Repeating the non-decay measurement with the period $t/n$, we
can describe the time evolution over the interval $[0,t]$ of a
state originally in the subspace $P\gotH$ by the interlaced
product $(P e^{-itH/n}P)^n$; the question is how this operator
will behave as $n\to\infty$. By QZE in the sense \cite{Fr} we mean
that such a limit will preserve the norm of the original state,
while quantum Zeno dynamics refers to the limiting operator itself
and its dependence upon the time $t$.
It is clear that there are Hamiltonians for which the question
does not make sense, a trivial example being constructed using the
momentum operator which generated a shift group in $L^2(\bR)$,
hence it is reasonable to assume that $H$ is semibounded (for a
physicist it means, of course, semibounded from below). However,
this assumption is not sufficient to justify the ``natural'' guess
about the limiting operator, namely that it might be equal to
$e^{-itH_P}$ with $H_P=PHP$; notice that in general the last
operator may not even be closed. Examples can be found -- see
\cite[Rem.~2.4.9]{Ex} or \cite{MaS} -- that the limit of $(P
e^{-itH/n}P)^n$ may not exist if the operator $PHP$ is ``too
small'', or more exactly,
if $\dom(H^{1/2}) \cap \goth$ is not dense in $\goth$. It appears thus to be more reasonable to replace it by
$H_P= (\sqrt{H}P)^*\sqrt{H}P$ assuming that the closed operator
$\sqrt{H}P$ is densely defined, then the question is in
which sense the limit of $(P e^{-itH/n}P)^n$ exists as
$n\to\infty$ and whether it equals $e^{-itH_P}$ with the last
named generator.
A partial answer has been given in \cite{EI} where the limit
existence is demonstrated for almost all $t$ in the strong
operator topology, along a subsequence $\{n'\}$ of natural
numbers\footnote{This fact was omitted in the first version of the
paper from which the claim was reproduced in the review
\cite{Sch}. A complete answer is at the present time known only in
the case when $P$ is finite-dimensional -- cf.~\cite{EI}.}, or
more generally, without taking a subsequence but with respect to a
slightly weaker topology which involves averaging in the parameter
$t$, see Remark~\ref{more on EI}. The reason behind this
restriction is that the exponential function involved in the
interlaced product gives rise to oscillations which are not easy
to deal with. One of the main ingredients in the present paper is
a simple observation that one can avoid the mentioned problem when
$f(s)= e^{-is}$ is replaced by other functions which will allow us
to use a positivity-type argument. We will illustrate this idea in
the next section on the case when the unitary group is replaced by
the resolvent, i.e. $f(s)= (1+is)^{-1}$. While such a formula is
not directly related to our physical motivation described above,
it represents a mathematical interest of its own\footnote{In
particular, our argument provides an easy way to show that for
resolvents in the recent result of \cite{Ca} one need not resort
to subsequences or to a weaker topology.}.
In Sec.~\ref{s: exp} we return to the exponential function and
show that the strong convergence can be demonstrated if we modify
the original problem setting by multiplying the unitary group with
a spectral projection of the Hamiltonian corresponding to a
suitable interval, with the sequence of this intervals expanding
as $n\to\infty$ -- cf.~Thm~\ref{III.3}. The question is whether
such a product formula has something in common with the Zeno
problem which assumes repeated non-decay measurements. The meaning
of the factors which enter the modified product is obvious: we
start with ascertaining that the state is in the unstable-system
subspace $\goth =P\gotH$ and let it evolve for the time $t/n$.
Then, before checking again that it has not decayed we perform an
additional yes-no experiment \cite{Ja} which consists of ``energy
filtering'', that is, we let the system pass if the measured
energy value falls into the interval $[0,\pi n/t)$. After the
non-decay measurement characterized by the projection $P$ we let
the system evolve again for the time $t/n$ and keep repeating the
whole procedure over the whole time interval $[0,t]$.
Let us mention that from the mathematical point of view there is a
relation between the Zeno product formula with an energy filtering
and the unitary Lie-Trotter product formula. In fact, the ideas to
employ a spectral cut-off together with the exponential function,
and to replace the unitary group by a the resolvent are not new:
they were used to derive a modification of the unitary Lie-Trotter
formula in \cite{Ich} and \cite{La1, La2}, respectively, both for
the form sum of two non-negative self-adjoint operators.
Notice now that the combination of the energy filtering and
non-decay measurement following immediately one after another can
be regarded as \emph{a single generalized measurement.} In fact, a
product of two, in general non-commuting\footnote{We are primarily
interested, of course, in the nontrivial case when the $P$ does
not commute with $H$, and thus also with the spectral projections
$E_H([0,\pi n/t))$.} projections represents the simplest
non-trivial example of generalized observables\footnote{Since the
spectral projections involved commute with the evolution operator,
one can also replace the product $PE_H([0,\pi t/n))$ in our
formul{\ae} by $E_H([0,\pi t/n))PE_H([0,\pi t/n))$. Such
generalized observables represented by symmetrized projection
products have been recently studied as \emph{almost sharp quantum
effects} -- cf.~\cite{AG}.} realized as positive maps of the
respective space of density matrices \cite[Sec.~2.1]{Da1}. Thus
our mathematical result corresponds to a modified Zeno situation
with such generalized measurements, which depend on $n$ and tend
to the standard non-decay yes-no experiment as $n\to\infty$.
In the last two sections we will show how the convergence
properties of such product formul{\ae} can be improved, in
particular, demonstrating more restrictive assumptions on the
operators involved which allow us to pass to the operator-norm
topology. We will also comment briefly on possible extensions of
the present results.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Resolvent case} \label{s: res}
Let $H = H^* \ge 0$ be a self-adjoint operator in a separable
Hilbert space $\gotH$ and let $P$ be an orthogonal projection onto a
subspace $\goth \subseteq \gotH$. The positivity assumption is
made only for convenience; the validity of the results will extend
easily to any semibounded $H$. We set
%
\be\la{1.1} F(\gt) := P(I + i\gt H)^{-1}P: \goth \longrightarrow
\goth\,, \quad \gt > 0\,, \ee
%
and
%
\be\la{1.2} S(\gt) := \frac{I_\goth - F(\gt)}{\gt}: \goth
\longrightarrow \goth\,, \quad \gt > 0\,, \ee
%
where $I_\goth$ is the identity operator in the subspace $\goth$.
In the following we use the notation $PXP$ for the restriction of an
operator $X$ acting in $\gotH$ to the
subspace $\goth$, i.e., we set $PXP := PX\!\upharpoonright\!\goth$.
Let us assume that
%
\be\la{1.3}
\dom(T) := \dom(\sqrt{H}) \cap \goth
\ee
%
is dense in $\goth$. We set $T: \goth \longrightarrow \gotH$,
%
\be\la{1.4}
Tf := \sqrt{H}f, \quad f \in \dom(T).
\ee
%
Since the operator $\sqrt{H}$ is closed the operator
$T$ is also closed. Let
%
%
\be\la{1.4a}
K := T^*T: \goth \longrightarrow \goth
\ee
%
%
which defines a non-negative self-adjoint operator in $\goth$.
Notice that $K$ corresponds to the sesquilinear form $\gotk$,
%
\be\la{1.5} \gotk(f,g) = (\sqrt{H}f,\sqrt{H}g)\,, \quad f,g \in
\dom(T)\,. \ee
%
\br
%
{\rm This definition allows us to write the operator $H_P = (\sqrt{H}P)^*\sqrt{H}P$
mentioned in the introduction as $K\oplus 0$ and its exponential
as $e^{-itK} \oplus I_{\goth^\perp}$. We will concentrate on the
nontrivial part of the problem only and formulate the claims below
for convergence in $\goth$.}
%
\er
%
>From (\ref{1.2}) on gets in a straightforward way the
representation
%
\be\la{1.9} S(\gt) = \gt P\frac{H^2}{I + \gt^2H^2}P + iP\frac{H}{I
+ \gt^2H^2}P\,, \ee
%
hence
%
\be\la{1.10} iI_\goth + S(\gt) = iI_\goth + \gt P\frac{H^2}{I +
\gt^2H^2}P + iP\frac{H}{I + \gt^2H^2}P\,. \ee
%
Then obviously we have the representation
%
\bea\la{1.13} \lefteqn{ iI_\goth + S(\gt) = \left(I_\goth
+P\frac{H}{I +
\gt^2H^2}P\right)^{1/2} \times}\\
& & \times \left\{ iI_\goth + M(\gt)\right\} \left(I_\goth
+P\frac{H}{I + \gt^2H^2}P\right)^{1/2}\!, \quad \gt > 0\,,
\nonumber \eea
%
where we put
%
\bead
\lefteqn{
M(\gt) := \gt\left(I_\goth +P\frac{H}{I +
\gt^2 H^2}P\right)^{-1/2}\times}\\
& & \times \,P\frac{H^2}{I + \gt^2 H^2}P \left(I_\goth
+P\frac{H}{I + \gt^2 H^2}P\right)^{-1/2}\!, \quad \gt > 0\,. \eead
%
The relation (\ref{1.13}) yields
%
\bea\la{1.14}
\lefteqn{
(iI_\goth + S(\gt))^{-1} = \left(I_\goth +P\frac{H}{I +
\gt^2H^2}P\right)^{-1/2} \times}\\
& & \times \left\{ iI_\goth + M(\gt)\right\}^{-1} \left(I_\goth
+P\frac{H}{I + \gt^2H^2}P\right)^{-1/2}\!, \quad \gt > 0\,.
\nonumber \eea
%
\bl\la{II.1}
%
Let $H$ be a non-negative self-adjoint operator in $\gotH$ and
let $\goth$ be a subspace of $\gotH$. If $\dom(\sqrt{H}) \cap
\goth$ is dense in $\goth$, then
%
\be\la{1.15} \slim_{\gt\to 0}\left(I_\goth +P\frac{H}{I +
\gt^2H^2}P\right)^{-1} = (I_\goth + K)^{-1}. \ee
%
\el
%
\begin{proof}
%
Let us introduce a family of sesquilinear forms:
%
\bed \gotk_\gt(f,g) = \left(\left(I_\goth +P\frac{H}{I +
\gt^2H^2}P\right)f,g\right), \quad f,g \in \goth\,, \quad \gt >
0\,. \eed
%
A straightforward computation shows that sequence is monotone,
%
\bed \gotk_{\gt_1}(f,f) \le \gotk_{\gt_2}(f,f)\,, \quad 0 < \gt_2
\le \gt_1\,, \quad f \in \goth\,. \eed
%
Since $\gotk_\gt \le \gotk$ for $\gt>0$, and $\lim_{\gt\to
0}\gotk_\gt(f,f) = \gotk(f,f)$, $f \in \dom(\gotk)$, the assertion
expressed by \eqref{1.15} follows from Theorem VIII.3.13 of
\cite{Ka}.
%
\end{proof}
%
Further we need the following general but simple statement.
%
\bl\la{II.1a}
%
Let $\left\{X(\gt)\right\}_{\gt > 0}$, $\left\{Y(\gt)\right\}_{\gt
> 0}$, and $\left\{A(\gt)\right\}_{\gt > 0}$ be families of bounded
non-negative self-adjoint operators in $\goth$ such that the
condition
%
\bed 0 \le X(\gt) \le A(\gt) \le Y(\gt)\,, \quad \gt > 0\,, \eed
%
is satisfied. If $s\!-\!\lim_{\gt \to 0}X(\gt) = s\!-\!\lim_{\gt \to 0}Y(\gt) =
A$, where $A$ is a bounded self-adjoint operator in $\goth$, then
$s\!-\!\lim_{\gt \to 0}A(\gt) = A$.
%
\el
%
\begin{proof}
%
Since for each $f \in \goth$ we have
%
\bed (X(\gt)f,f) \le (A(\gt)f,f) \le (Y(\gt)f,f)\,, \quad \gt >
0\,, \eed
%
we get $\lim_{\gt\to 0}(A(\gt)f,f) = (Af,f)$, $f \in \goth$, or
$w\!-\!\lim_{\gt\to 0}A(\gt) = A$. By the L\"owner-Heinz
inequality \cite{BS} we find
%
\bed 0 \le X(\gt)^{1/2} \le A(\gt)^{1/2} \le Y(\gt)^{1/2}, \quad
\gt > 0\,. \eed
%
Since $s\!-\!\lim_{\gt\to 0}X(\gt)^{1/2}=s\!-\!\lim_{\gt\to
0}Y(\gt)^{1/2}=A^{1/2}$, we obtain the weak convergence
$w\!-\!\lim_{\gt\to 0}A(\gt)^{1/2}=A^{1/2}$. By
%
\bed
\left\|A(\gt)^{1/2}f - A^{1/2}f\right\|^2 = (A(\gt)f,f) - 2\real(A(\gt)^{1/2}f,A^{1/2}f) + (Af,f)
\eed
%
we get $\slim_{\gt\to 0}A(\gt)^{1/2} = A^{1/2}$, which yields
$\slim_{\gt \to 0}A(\gt) = A$.
%
\end{proof}
Let us now consider the operator family
%
\be\la{1.18} L(\gt) := P\frac{H}{I + \gt^2H^2}P + \gt
P\frac{H^2}{I + \gt^2H^2}P: \goth \longrightarrow \goth\,, \quad
\gt > 0\,, \ee
%
for which we can prove a claim analogous to Lemma~\ref{II.1}.
%
\bl\la{II.5}
%
Under the assumptions of Lemma~\ref{II.1} we have
%
\be\la{1.19} \slim_{\gt\to 0} (I_\goth + L(\gt))^{-1} = (I_\goth +
K)^{-1}. \ee
%
\el
%
\begin{proof}
%
First we note that
%
\be\la{1.20} (I_\goth + L(\gt))^{-1} \le \left(I_\goth +
P\frac{H}{I + \gt^2H^2}P\right)^{-1}, \quad \gt > 0\,. \ee
%
Since
%
\bed I_\goth + P\frac{H}{I + \gt^2H^2}P \le I_\goth + K\,, \quad
\gt > 0\,, \eed
%
we find
%
\bed I_\goth + P\frac{H}{I + \gt^2H^2}P + \gt P\frac{H^2}{I +
\gt^2H^2}P \le I_\goth + K + \gt P\frac{H^2}{I + \gt^2H^2}P\,,
\quad \gt > 0\,, \eed
%
which in turn yields
%
\bed \left(I_\goth + K + \gt P\frac{H^2}{I +
\gt^2H^2}P\right)^{-1} \le (I_\goth + L(\gt))^{-1}, \quad \gt >
0\,. \eed
%
Hence we get a two-sided estimate,
%
\bea\la{1.20a}
\lefteqn{
\left(I_\goth + K + \gt P\frac{H^2}{I +
\gt^2H^2}P\right)^{-1} \le }\\
& & \le (I_\goth + L(\gt))^{-1} \le \left(I_\goth + P\frac{H}{I +
\gt^2H^2}P\right)^{-1}, \quad \gt > 0\,, \nonumber \eea
%
where the left-hand side can be represented as follows,
%
\bea\la{1.21}
\lefteqn{
\left(I_\goth + K + \gt P\frac{H^2}{I + \gt^2H^2}P\right)^{-1}
= (I_\goth + K)^{-1/2}\,\times}\\
& &
\times \left(I_\goth + \gt (I_\goth + K)^{-1/2}P\frac{H^2}{I + \gt^2H^2}P(I_\goth +
K)^{-1/2}\right)^{-1}\times
\nonumber\\
& & \times\, (I_\goth + K)^{-1/2}, \quad \gt > 0\,.\nonumber \eea
%
By virtue of \eqref{1.5} we infer that
%
\bead \lefteqn{ (\sqrt{H}P(I_\goth + K)^{-1/2}f,\sqrt{H}P(I_\goth
+ K)^{-1/2}f) = }\\
& & = \gotk((I_\goth + K)^{-1/2}f,(I_\goth + K)^{-1/2}f) =
\|\sqrt{K}(I_\goth + K)^{-1/2}f\|^2 \le \|f\|^2, \eead
%
for any $f \in \goth$, which yields
%
\bed \|W f\| \le \|f\|\,, \quad f \in \goth\,, \eed
%
where $W := \sqrt{H}P(I_\goth + K)^{-1/2}$. From the identity
%
\bead
\lefteqn{
\left(\frac{\gt H}{I + \gt^2H^2}Wf,Wg\right) =}\\
& & \left(\frac{\gt H}{I + \gt^2H^2}\sqrt{H}P(I_\goth +
K)^{-1/2}f,\sqrt{H}P(I_\goth + K)^{-1/2}g\right) \eead
%
we obtain
%
\bed \gt (I_\goth + K)^{-1/2}P\frac{H^2}{I + \gt^2H^2}P(I_\goth +
K)^{-1/2} = W^*\frac{\gt H}{I + \gt^2H^2}W\,. \eed
%
Since
%
\bed \slim_{\gt\to 0} \frac{\gt H}{1 + \gt^2 H^2} = 0 ,\eed
%
and $W$ is bounded we conclude that
%
\bed \slim_{\gt\to 0}(I_\goth + K)^{-1/2}P\frac{\gt H^2}{I +
\gt^2H^2}P(I_\goth + K)^{-1/2} = 0\,. \eed
%
Using the representation \eqref{1.21} we arrive at
%
\bed \slim_{\gt\to 0} \left(I_\goth + K + \gt P\frac{H^2}{I +
\gt^2H^2}P\right)^{-1} = (I_\goth + K)^{-1}\,; \eed
%
taking now into account the inequalities \eqref{1.20a},
Lemma~\ref{II.1} and applying Lemma \ref{II.1a} we complete the
proof.
%
\end{proof}
%
Now we are in position to prove the main result of this section.
%
\bt\la{II.6}
%
Let $H$ be a non-negative self-adjoint operator in $\gotH$ and let
$\goth$ be a closed subspace of $\gotH$. If $\dom(\sqrt{H}) \cap
\goth$ is dense in $\goth$, then we have for any $t_0> 0$ the
limit
%
\bed \slim_{n\to\infty} \left(P(I + itH/n)^{-1}P\right)^n =
e^{-itK}\,, \eed
%
uniformly in $t \in [0,t_0]$, where the generator $K$ is defined
by (\ref{1.5}).
%
\et
%
\begin{proof}
%
Since
%
\bead
\lefteqn{
(I_\goth + L(\gt))^{-1} = }\\
& & \left(I_\goth +P\frac{H}{I + \gt^2H^2}P\right)^{-1/2} (I +
M(\gt))^{-1} \left(I_\goth +P\frac{H}{I + \gt^2H^2}P\right)^{-1/2}
\eead
%
holds for any $\gt > 0$, by Lemmata \ref{II.1} and \ref{II.5} we
infer that
%
\bed \wlim_{\gt\to 0}(I_\goth + M(\gt))^{-1} = I_\goth\,, \eed
%
which yields
%
\bed \slim_{\gt\to 0}\left(I_{\goth} - (I_\goth +
M(\gt))^{-1}\right)^{1/2} = 0 \,.\eed
%
Hence the above limit is strong at the same time,
%
\bed \slim_{\gt\to 0} (I_\goth + M(\gt))^{-1} = I_\goth \,. \eed
%
By the resolvent identity this gives
%
\bed \slim_{\gt\to 0} (iI_\goth + M(\gt))^{-1} = -iI_\goth \,;
\eed
%
then using the representation \eqref{1.14} and applying Lemma
\ref{II.1} we find
%
\bed \slim_{\gt\to 0} (iI_\goth + S(\gt))^{-1} = -i(I_\goth +
K)^{-1} \eed
%
or, by the resolvent identity again,
%
\bed \slim_{\gt\to 0} (I_\goth + S(\gt))^{-1} = (I_\goth +
iK)^{-1} \,. \eed
%
The sought result then follows from Chernoff's theorem \cite{Ch},
see also Lemma~3.29 of \cite{Da2}.
%
%
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Exponential case} \label{s: exp}
%
Let us now pass to the second problem described in the
introduction. We shall consider the operator family
%
\bed F(\gt) := PE_H(\gD_\gt)e^{-i\gt H}P: \goth \longrightarrow
\goth\,, \quad \gt > 0 \,, \eed
%
where $E_H (\cdot)$ is the spectral measure corresponding to $H$
and
%
\bed \gD_\gt := [0,\pi/\gt)\,. \eed
%
Furthermore, in analogy with (\ref{1.2}) we set
%
\bed S(\gt) := \frac{I_\goth - F(\gt)}{\gt}: \goth \longrightarrow
\goth\,, \quad \gt > 0\,. \eed
%
This operator family has the representation
%
\bed S(\gt) = \frac{1}{\gt}P(I - E_H(\gD_\gt) \cos(\gt H))P +
i\frac{1}{\gt}PE_H(\gD_\gt)\sin(\gt H)P\,. \eed
%
By the factorization
%
\bead
\lefteqn{
iI_\goth + S(\gt) = \left(I_\goth + \frac{1}
{\gt}PE_H(\gD_\gt)\sin(\gt H)P\right)^{1/2} \times}\\
& & \times \left\{iI_\goth + M(\gt)\right\}\left(I_\goth +
\frac{1}{\gt}PE_H(\gD_\gt)\sin(\gt H)P\right)^{1/2}\!, \eead
%
where
%
\bead
\lefteqn{
M(\gt):= \left(I_\goth + \frac{1}{\gt}PE_H(\gD_\gt)
\sin(\gt H)P\right)^{-1/2} \times}\\
& & \times \frac{1}{\gt}P(I - E_H(\gD_\gt) \cos(\gt H))P
\left(I_\goth +\frac{1}{\gt}PE_H(\gD_\gt)\sin(\gt
H)P\right)^{-1/2}\!, \eead
%
one gets
%
\bea\la{3.0a}
\lefteqn{
(iI_\goth + S(\gt))^{-1} = \left(I_\goth +
\frac{1}{\gt}PE_H(\gD_\gt)\sin(\gt H)P\right)^{-1/2}
\times}\\
& & \times \left\{iI_\goth + M(\gt)\right\}^{-1} \left(I_\goth +
\frac{1}{\gt}PE_H(\gD_\gt)\sin(\gt H)P\right)^{-1/2}. \nonumber
\eea
%
Recall that a piecewise smooth\footnote{This notion appears in the
literature with different regularity requirements; the present
form is sufficient for purposes of the present paper -- see
\cite[Chap.~4]{Za}.} function $f:\,\bR_+\to [0,1]$ is called a
\emph{Kato function} if $f(0)=1$ and $f'(0+)=-1$.
%
\bl\la{III.1}
%
Let $H$ be a non-negative self-adjoint operator in $\gotH$ and let
$\goth$ a subspace of $\gotH$. If $\dom(\sqrt{H}) \cap \goth$ is
dense in $\goth$ and a Kato function $g$ satisfies the condition
%
\bed
\sup_{\gl > 0}\frac{1 - g(\gl)}{\gl} \le 1\,,
\eed
%
then we have
%
\be\la{3.00} \slim_{\gt\to 0}\left(I_\goth + \frac{1}{\gt}P(I -
g(\gt H))P\right)^{-1} = (I_\goth + K)^{-1}. \ee
%
\el
%
\begin{proof}
%
By assumption of the function $g$ we have
%
\bed \left(\frac{1}{\gt}P(I - g(\gt H))Pf,f\right) \le
(\sqrt{H}Pf,\sqrt{H}Pf)\,, \quad f \in \dom(T)\,, \eed
%
or equivalently
%
\bed \frac{1}{\gt}P(I - g(\gt H))P \le K\,, \quad \gt > 0\,, \eed
%
which implies
%
\be\la{3.0} (I_\goth + K)^{-1} \le \left(I_\goth +
\frac{1}{\gt}P(I - g(\gt H))P\right)^{-1}\,, \quad \gt > 0\,. \ee
%
Furthermore, we have
%
\bed \frac{1}{\gt}P(I - g(\gt H))E_H([0,a))P \le \frac{1}{\gt}P(I
- g(\gt H))P\,, \quad a > 0\,, \eed
%
which shows that
%
\bea\la{3.1}
\lefteqn{
\left(I_\goth + \frac{1}{\gt}P(I - g(\gt H))P\right)^{-1} \le}\\
& & \times \left(I_\goth + \frac{1}{\gt}P(I - g(\gt
H))E_H([0,a))P\right)^{-1}\,, \quad a > 0\,. \nonumber \eea
%
It is obvious that
%
\bed \slim_{\gt\to 0} \frac{1}{\gt}P(I - g(\gt H))E_H([0,a))P =
PHE_H([0,a))P\,, \quad a > 0\,, \eed
%
and moreover,
%
\be\la{3.1a} \slim_{\gt \to 0}\left(I_\goth + \frac{1}{\gt}P(I -
g(\gt H))E_H([0,a))P\right)^{-1} = \left(I_\goth +
PHE_H([0,a))P\right)^{-1} \ee
%
holds for all $a > 0$ by Chernoff's theorem \cite{Ch}, see also
Lemma~3.29 of \cite{Da2}. Since $\{PHE_H([0,a))P\}_{a > 0}$, is a
monotone increasing family of operators, which obeys
$PHE_H([0,a))P \le K$ and
%
\bed \lim_{a\to\infty}(E_H([0,a))\sqrt{H}f,\sqrt{H}f) =
(\sqrt{H}f,\sqrt{H}f)\,, \quad f\in \dom(T)\,, \eed
%
by Theorem VIII.3.13 of \cite{Ka} we get that
%
\bed \slim_{a\to\infty}\left(I_\goth + PHE_H([0,a))P\right)^{-1} =
(I_\goth + K)^{-1}. \eed
%
>From \eqref{3.1} and \eqref{3.1a} we obtain
%
\bead
\lefteqn{
\lim\sup_{\gt\to 0}\left(\left(I_\goth + \frac{1}
{\gt}P(I - g(\gt H))P\right)^{-1}f,f\right) \le}\\
& & \le \left(\left(I_\goth +
PHE_H([0,a))P\right)^{-1}f,f\right)\,, \quad a > 0, \quad f \in
\goth\,, \eead
%
which yields
%
\bed \lim\sup_{\gt\to 0}\left(\left(I_\goth + \frac{1}{\gt}P(I -
g(\gt H))P\right)^{-1}f,f\right) \le \left(\left(I_\goth +
K\right)^{-1}f,f\right)\,, \quad f \in \goth\,. \eed
%
Using \eqref{3.0} we obtain
%
\bed \left(\left(I_\goth + K\right)^{-1}f,f\right) \le
\lim\inf_{\gt\to 0}\left(\left(I_\goth + \frac{1}{\gt}P(I - g(\gt
H))P\right)^{-1}f,f\right)\,, \quad f \in \goth\,. \eed
%
and therefore
%
\bed \lim_{\gt\to 0}\left(\left(I_\goth + \frac{1}{\gt}P(I - g(\gt
H))P\right)^{-1}f,f\right) = \left(\left(I_\goth +
K\right)^{-1}f,f\right)\,, \quad f \in \goth\,. \eed
%
This yields
%
\be\la{3.2} \wlim_{\gt\to 0}\left(I_\goth + \frac{1}{\gt}P(I -
g(\gt H))P\right)^{-1} = \left(I_\goth + K\right)^{-1}. \ee
%
Using once more the L\"owner-Heinz inequality and taking into
account \eqref{3.0} and \eqref{3.1} we infer that
%
\bead
\lefteqn{
(I_\goth + K)^{-1/2} \le \left(I_\goth + \frac{1}
{\gt}P(I - g(\gt H))P\right)^{-1/2} \le}\\
& & \le \left(I_\goth + \frac{1}{\gt}P(I - g(\gt
H))E_H([0,a))P\right)^{-1/2}, \quad a > 0\,, \quad \gt > 0\,.
\eead
%
Mimicking the reasoning of the previous section we get
%
\be\la{3.3} \wlim_{\gt\to 0}\left(I_\goth + \frac{1}{\gt}P(I -
g(\gt H))P\right)^{-1/2} = \left(I_\goth + K\right)^{-1/2}, \ee
%
and \eqref{3.2} in combination with \eqref{3.3} finally yield the
relation \eqref{3.00}
%
\end{proof}
Now we set
%
\bed g(\gl) := \left\{ \ba{lcl}
1- \sin(\gl) \quad&\dots&\quad \gl \in [0,\pi)\\
1 \quad&\dots&\quad \gl \in [\pi,\infty) \ea \right. \eed
%
One easily verifies that $g(\cdot)$ is a Kato function, and by
functional calculus
%
\bed I - g(\gt H) = \sin(\gt H)E_H(\gD_\gt)\,. \eed
%
Furthermore, we have
%
\bed \sup_{\gl > 0}\frac{1 - g(\gl)}{\gl} = \sup_{\gl >
0}\frac{\sin(\gl)}{\gl}\chi_{[0,\pi)}(\gl) \le 1\,. \eed
%
This allows us to apply Lemma \ref{III.1} which yields
%
\be\la{3.3c} \slim\left(I_\goth + \frac{1}{\gt}PE_H(\gD_\gt)
\sin(\gt H)P\right)^{-1} = (I_\goth + K)^{-1}. \ee
%
Next we set
%
\bed L(\gt) := \frac{1}{\gt}PE_H(\gD_\gt)\sin(\gt H)P +
\frac{1}{\gt}P(I - E_H(\gD_\gt) \cos(\gt H))P \,.\eed
%
\bl\la{III.2}
%
Under the assumptions of Lemma~\ref{II.1} we have
%
\be\la{3.3d} \slim_{\gt\to 0} (I_\goth + L(\gt))^{-1} = (I_\goth +
K)^{-1}. \ee
%
\el
%
\begin{proof}
%
We note that
%
\be\la{3.3a}
(I_\goth + L(\gt))^{-1} \le \left(I_\goth + \frac{1}
{\gt}PE_H(\gD_\gt)\sin(\gt H)P\right)^{-1}.
\ee
%
Since
%
\bed \frac{1}{\gt}PE_H(\gD_\gt)\sin(\gt H)P \le K \,,\eed
%
we have also a lower to the left-hand side of \eqref{3.3a},
%
\be\la{3.3b} (I_\goth + K + \frac{1}{\gt}P(I - E_H(\gD_\gt)
\cos(\gt H))P)^{-1} \le (I_\goth + L(\gt))^{-1}. \ee
%
Now we use the representation
%
\bea\la{3.4}
\lefteqn{
(I_\goth + K + \frac{1}{\gt}P(I - E_H(\gD_\gt)
\cos(\gt H))P)^{-1} =}\\
& & =(I_\goth + K)^{-1/2}(I_\goth + N(\gt))^{-1}(I_\goth +
K)^{-1/2}\,, \nonumber \eea
%
where
%
\bed N(\gt) := \frac{1}{\gt} (I_\goth + K)^{-1/2}P(I -
E_H(\gD_\gt) \cos(\gt H))P(I_\goth + K)^{-1/2}. \eed
%
This operator can be rewritten in the form
%
\bead
\lefteqn{
N(\gt) = \frac{1}{\gt} (I_\goth + K)^{-1/2}
PE_H([\pi/\gt,\infty))P(I_\goth + K)^{-1/2} +}\\
& & +\frac{1}{\gt} (I_\goth + K)^{-1/2}PE_H(\gD_\gt)(I - \cos(\gt
H))P(I_\goth + K)^{-1/2}, \eead
%
%
and using the contraction operator $W := \sqrt{H}P(I_\goth +
K)^{-1/2}$ we get
%
\bed \frac{1}{\gt} (I_\goth +
K)^{-1/2}PE_H([\pi/\gt,\infty))P(I_\goth + K)^{-1/2} = W^*(\gt
H)^{-1}E_H([\pi/\gt,\infty))W \,. \eed
%
Now since $\|(\gt H)^{-1}E_H([\pi/\gt,\infty))\| \le \frac{1}{\pi}
\le 1$, we obtain the estimate
%
\bed \frac{1}{\gt} \|(I_\goth +
K)^{-1/2}PE_H([\pi/\gt,\infty))P(I_\goth + K)^{-1/2}f\| \le
\|E_H([\pi/\gt,\infty))Wf\|\,, \eed
%
which yields
%
\bead \lefteqn{ \lim_{\gt\to 0}\frac{1}{\gt} \|(I_\goth +
K)^{-1/2}
PE_H([\pi/\gt,\infty))P(I_\goth + K)^{-1/2}f\|} \\
& & \le \lim_{\gt\to 0}\|E_H([\pi/\gt,\infty))Wf\| = 0 \,. \eead
%
Moreover, by virtue of the limit
%
\bed \slim_{\gt\to 0}\frac{1}{\gt}(I - \cos(\gt H))f = 0\,, \quad
f \in \dom(H)\,, \eed
%
we infer that
%
\bed \slim_{\gt\to 0}\frac{1}{\gt} (I_\goth +
K)^{-1/2}PE_H(\gD_\gt)(I - \cos(\gt H))P(I_\goth + K)^{-1/2} =
0\,, \eed
%
hence
%
\bed \slim_{\gt\to 0}N(\gt) = 0\,. \eed
%
%
Taking now into account the representation \eqref{3.4} we find
%
\bed \slim_{\gt \to 0}(I_\goth + K + \frac{1}{\gt}P(I -
E_H(\gD_\gt) \cos(\gt H))P)^{-1} = (I_\goth + K)^{-1}\,; \eed
%
from \eqref{3.3a} and \eqref{3.3b} in combination with
\eqref{3.3c} and Lemma \ref{II.1a} we conclude finally that
\eqref{3.3d} is valid.
%
\end{proof}
%
\bt\la{III.3}
%
Let $H$ be a non-negative self-adjoint operator on
$\gotH$ and let $\goth$ be a closed subspace of
$\gotH$. If the set $\dom(\sqrt{H}) \cap \goth$ is dense in $\goth$,
then
%
\bed \slim_{n\to\infty} \left(PE_H([0,\pi
n/t))e^{-itH/n}P\right)^n = e^{-itK} \eed
%
uniformly in $t \in [0,t_0]$ for any $t_0 > 0$, where $K$ is
defined by (\ref{1.5}).
%
\et
%
\begin{proof}
%
Since
%
\bead
\lefteqn{
(I_\goth + L(\gt))^{-1} = \left(I_\goth +
\frac{1}{\gt}PE_H(\gD_\gt)\sin(\gt H)P\right)^{-1/2}\times}\\
& & \times\left\{I_\goth + M(\gt)\right\}^{-1}\left(I_\goth +
\frac{1}{\gt}PE_H(\gD_\gt)\sin(\gt H)P\right)^{-1/2} ,\eead
%
one can use \eqref{3.3c} and (\ref{3.3d}) to conclude that
%
\bed \wlim_{\gt\to 0}\left(I_\goth + M(\gt)\right)^{-1} = I_\goth
\,.\eed
%
Repeating the argument from proof of Theorem~\ref{II.6} we get
%
\bed \slim_{\gt\to 0}\left(I_\goth + M(\gt)\right)^{-1} = I_\goth
\,.\eed
%
Then \eqref{3.0a} and \eqref{3.3c} yield
%
\bed \slim_{\gt\to 0}\left(iI_\goth + S(\gt)\right)^{-1} =
-i(I_\goth + K)^{-1} ,\eed
%
or equivalently
%
\bed \slim_{\gt\to 0}\left(I_\goth - i S(\gt)\right)^{-1} =
(I_\goth + K)^{-1} . \eed
%
The last limit together with Chernoff's theorem \cite{Ch}, see
also Lemma 3.29 of \cite{Da2}, complete the proof in the
exponential case.
%
\end{proof}
%
%
\br {\em
%
%
The conclusion of this section can be extended to other families
of ``filter windows''. Inspecting the proof, one sees that
$\gD_\gt := [0,\pi/\gt)$ can be replaced by any family
$\{\tilde\gD_\gt \}$ such that $\tilde\gD_\gt \subset \gD_\gt$ and
$\tilde\gD_\gt\to \bR_+$ as $\gt\to 0+$.
%
%
} \er
%
% ------------- %
\br \label{more on EI}
{\em The result given in Theorem~\ref{III.3} is in a sense
complementary to the conclusions of \cite{EI} where the formula
without the spectral projection $E_H([0,\pi n/t))$ was proved,
however, the convergence was demonstrated only in a weaker
topology, namely that of $L_\mathrm{loc}^2(\bR; {\cal H})$. Recall
that it implies existence of a set $M \subset \bR$ of Lebesgue
measure zero and a strictly increasing sequence $\{n'\}$ of
positive integers along which $(Pe^{-itH/n'}P)^{n'} \to e^{-itK}$
holds strongly for all $t \in \bR \setminus M$. A complete result without resorting to a
subsequence and exclusion of a zero-measure set still waits for
its proof. }
\er
% ------------- %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Operator norm convergence} \label{s: norm}
In the following we are going to ask how the convergence
properties of the above derived product formula can be improved
under more restrictive assumptions assumptions. To this end we set
%
%
\bed F_0(\gt) := Pe^{-i\gt H}P: \goth \longrightarrow \goth\,,
\quad \gt > 0\,, \eed
%
%
and
%
%
\bed S_0(\gt) := \frac{I_\goth - F_0(\gt)}{\gt}: \goth
\longrightarrow \goth\,, \quad \gt > 0\,. \eed
%
We will show first that if $\sqrt{H}$ is defined everywhere on
$\goth$ one can get rid of the spectral projections used in the
previous section.
%
\bt\la{IV.1}
%
%
Let $H$ be a non-negative self-adjoint operator on
$\gotH$ and let $\goth$ be a closed subspace of
$\gotH$. If $\goth \subseteq \dom(\sqrt{H})$, then
%
\be\la{4.1} \slim_{n\to\infty} \left(Pe^{-itH/n}P\right)^n =
e^{-itK} \ee
%
uniformly in $t \in [0,t_0]$ for any $t_0 > 0$ where $K$ is
defined by (\ref{1.5}).
%
\et
%
%
\begin{proof}
%
%
To begin with, note that $\goth \subseteq \dom(\sqrt{H})$ implies
that $T = \sqrt{H}P$ is a bounded operator, and consequently, $K =
T^*T$ is also bounded. Then we may employ the representation
%
%
\be\la{4.1a} (I_\goth - F_0(\gt))f = i\int^\gt_0
ds\;T^*e^{-isH}Tf\,, \quad \gt > 0\,, \quad f \in \goth\,, \ee
%
%
or
%
%
\bed S_0(\gt)f - iKf = \frac{i}{\gt}\int^\gt_0 ds \;
T^*\left(e^{-isH} - I_\goth\right)Tf\,, \quad \gt > 0\,, \quad f
\in \goth\,, \eed
%
%
which provides us with the estimate
%
%
\bed \|S_0(\gt)f - iKf\|_\goth \le \sup_{s \in
[0,\gt]}\|T^*\left(e^{-isH} - I_\goth\right)Tf\|_\goth \quad \gt >
0\,, \quad f \in \goth\,. \eed
%
%
Since
%
%
\bed \lim_{\gt\to 0} \sup_{s \in [0,\gt]}\|T^*\left(e^{-isH} -
I_\goth\right)Tf\|_\goth = 0\,, \quad f \in \goth\,, \eed
%
%
we immediately conclude that
%
%
\bed \slim_{\gt\to 0} S_0(\gt) = iT^*T = iK\,. \eed
%
%
In this way we obtain the relation
%
%
\bed \slim_{\gt\to 0}(I_\goth + S_0(\gt))^{-1} = (I_\goth +
iK)^{-1}, \eed
%
%
and using Chernoff's theorem \cite{Ch} one more time we prove
(\ref{4.1}).
%
%
\end{proof}
%
%
\bc\la{IV.2}
%
%
Let $H$ be a non-negative self-adjoint operator in $\gotH$ and let
$\goth$ be a closed subspace of $\gotH$ such that $\goth \subseteq
\dom(\sqrt{H})$. If in addition the operator $T$ is compact or
$\goth \subseteq \dom(H^\ga)$ is satisfied for some $\ga \in
(\frac12,1]$, then
%
\be\la{4.2}
\lim_{n\to\infty} \left\|\left(Pe^{-itH/n}P\right)^n
-e^{-itK}\right\|_{\kB(\goth)} = 0
\ee
%
holds uniformly in $t \in [0,t_0]$ for any $t_0 > 0$.
%
%
\ec
%
\begin{proof}
%
%
Since
%
%
\bed S_0(\gt)f - iKf = \frac{i}{\gt}\int^\gt_0 ds\;
T^*\left(e^{-isH} - I_\goth\right)Tf\,, \quad \gt > 0\,, \quad f
\in \goth\,, \eed
%
%
we can write down the estimate
%
%
\be\la{4.3} \left\|S_0(\gt) - iK\right\|_{\kB(\gotH)} \le \sup_{s
\in [0,\gt]}\left\|T^*\left(e^{-isH} -
I_\goth\right)T\right\|_{\kB(\gotH)}\,, \quad \gt > 0\,. \ee
%
%
If $T$ is compact, then
%
%
\bed \lim_{\gt\to 0}\sup_{s \in [0,\gt]}\left\|T^*\left(e^{-isH} -
I_\goth\right)T\right\|_{\kB(\gotH)} = 0\,; \eed
%
%
the argument is analogous to that in the proof of Proposition~2.69
in \cite{Za}. In combination with \eqref{4.3} this yields
%
%
\be\la{4.4}
\lim_{\gt \to 0}\left\|S_0(\gt) - iK\right\|_{\kB(\gotH)} = 0.
\ee
%
%
If $\goth \subseteq \dom(H^\ga)$ for some $\ga \in (\frac12,1]$,
then we set $T_\ga := H^\ga P$ and $K_\ga = T^*_\ga T_\ga$. Notice
that $T_\ga$ is a bounded operator. From (\ref{4.3}) we obtain the
estimate
%
%
\bed \left\|S_0(\gt) - iK\right\|_{\kB(\gotH)} \le \sup_{s \in
[0,\gt]}s^{2\ga-1} \left\|T^*_\ga\;\frac{e^{-isH} -
I_\goth}{(sH)^{2\ga-1}}\;T_\ga\right\|_{\kB(\gotH)}\,, \quad \gt >
0\,. \eed
%
%
Using the inequality
%
%
\bed \sup_{\gl \ge 0}\left|\frac{e^{-i\gl} -
1}{\gl^{2\ga-1}}\right| \le 2^{2(1-\ga)} \eed
%
%
we find
%
%
\bed \left\|S_0(\gt) - iK\right\|_{\kB(\gotH)} \le
2^{2(1-\ga)}\gt^{2\ga-1}\|K_\ga\|_{\kB(\goth)}\,, \quad \gt > 0\,,
\eed
%
%
which yields again the relation (\ref{4.4}).
Next we use the expansions of the operator families in question,
%
%
\bed e^{-tS_0(t/n)} - e^{-itK} = \sum^\infty_{j=0}
\frac{(-t)^k}{j!}\left(S_0(t/n)^j - (iK)^j\right)\,, \eed
%
%
which together with (\ref{4.4}) allows us to conclude that
%
%
\be\la{4.5}
\lim_{n\to\infty}\left\|e^{-tS_0(t/n)} - e^{-itK}\right\|_{\kB(\goth)}
=0
\ee
%
%
holds for any $t > 0$, uniformly in $t \in [0,t_0]$. We shall
combine it with the telescopic estimate
%
%
\bea\la{4.5a}
\lefteqn{
\left\|F_0(t/n)^n - e^{-itK}\right\|_{\kB(\goth)} \le}\\
& & \le \left\|F_0(t/n)^n - e^{-tS_0(t/n)}\right\|_{\kB(\goth)} +
\left\|e^{-tS_0(t/n)} - e^{-itK}\right\|_{\kB(\goth)}\,,\nonumber
\eea
%
%
where the first term can be treated as in Lemma 3.27 of
\cite{Da2},
%
%
\be\la{4.6} \left\|\left(F_0(t/n)^n -
e^{-tS_0(t/n)}\right)f\right\|_\goth \le
\sqrt{n}\left\|\left(F_0(t/n) - I_\goth\right)f\right\|_\goth\,,
\quad f \in \goth\,. \ee
%
%
Using the representation (\ref{4.1a}) with $\gt = t/n$, we can
estimate the right-hand side of \eqref{4.6} by
%
%
\bed \left\|\left(F_0(t/n) - I_\goth\right)f\right\|_\goth \le
\int^{t/n}_0 ds \; \|T^* e^{-isH} Tf\|_\goth\,, \quad f \in
\goth\,, \eed
%
%
which gives
%
%
\bed \left\|\left(F_0(t/n) - I_\goth\right)f\right\|_\goth \le
\frac{t}{n} \|K\|_{\kB(\goth)}\|f\|\,, \quad f \in \goth\,. \eed
%
%
Inserting this estimate into (\ref{4.6}) we obtain
%
%
\bed
\left\|F_0(t/n)^n - e^{-tS_0(t/n)}\right\|_{\kB(\goth)} \le
\frac{t}{\sqrt{n}}\|K\|_{\kB(\goth)}
\eed
%
%
which yields
%
%
\be\la{4.8}
\lim_{n\to\infty}\left\|F_0(t/n)^n -
e^{-tS_0(t/n)}\right\|_{\kB(\goth)} = 0
\ee
%
%
for any $t > 0$, uniformly in $t \in [0,t_0]$. Taking into account
(\ref{4.5}), (\ref{4.5a}) and (\ref{4.8}) we arrive at the sought
relation (\ref{4.2}).
%
%
\end{proof}
%
%
\br
{\em
%
%
Obviously, the conclusion (\ref{4.2}) is valid if $\goth \subseteq
\dom(\sqrt{H})$ and $\goth$ is a finite dimensional subspace.
Indeed, in this case the operator $T$ is finite dimensional, and
therefore compact. This gives an alternative proof of the result
derived in Sec.~5 of \cite{EI}.
%
%
}
\er
%
\br
%
{\em In connection with the last remark let us mention that in
the finite-dimensional case there is one more way to prove the
claim suggested by G.M.~Graf and A.~Guekos \cite{GG}. The argument
is based on the observation that
%
%
\be\la{4.99} \lim_{t\to 0}\: t^{-1}\left\| P e^{-itH}P - P
e^{-itK}P \right\|_{\kB(\goth)} = 0 \ee
%
implies $\left\| (P e^{-itH/n}P)^n - P e^{-itK}
\right\|_{\kB(\goth)} = n\, o(t/n)$ as $n\to\infty$ by means of a
natural telescopic estimate. To establish (\ref{4.99}) one first
proves that
%
\bed t^{-1} \left[ (f, P e^{-itH}Pg)_{\kB(\goth)} -
(f,g)_{\kB(\goth)} -it (\sqrt{H}Pf, \sqrt{H}Pg)_{\kB(\goth)}
\right] \longrightarrow 0 \eed
%
as $t\to 0$ for all $f,g$ from $\dom(\sqrt{H}P)$ which coincides
in this case with $\goth$ by the closed-graph theorem. The last
expression is equal to
%
\bed \left( \sqrt{H}Pf, \left[ \frac{e^{-itH}-I}{tH} -i \right]
\sqrt{H}Pg \right)_{\kB(\goth)} \eed
%
and the square bracket tends to zero strongly by the functional
calculus, which yields the sought conclusion. In the same way we
find that
%
\bed t^{-1} \left[ (f, P e^{-itK}Pg)_{\kB(\goth)} -
(f,g)_{\kB(\goth)} -it (\sqrt{K}f, \sqrt{K}g)_{\kB(\goth)} \right]
\longrightarrow 0 \eed
%
holds as $t\to 0$ for any vectors $f,g\in\goth$. Next we note that
$(\sqrt{K}f, \sqrt{K}g)_{\kB(\goth)} = (\sqrt{H}Pf,
\sqrt{H}Pg)_{\kB(\goth)}$, and consequently, the expression
contained in (\ref{4.99}) tends to zero weakly as $t\to 0$,
however, in a finite dimensional $\goth$ the weak and
operator-norm topologies are equivalent.
}
%
\er
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Concluding remarks}
In this paper we have been able to make a derive a pair of new
product formul{\ae} related to the Zeno problem. While the results
of Section~\ref{s: norm} contribute to the problem in its standard
form, important questions about the sense in which Zeno dynamics
exists remain open.
At the same time we demonstrated here that the problem can be
interpreted in a broader sense, with suitable generalized
observables replacing the usual non-decay measurement. We have
analyzed here the simplest nontrivial situation when we deal with
a product of two non-commuting projections. One can ask, of
course, what will happen if one considers instead more complicated
measurement characterized by positive-operator-valued measures
(POVM), as analyzed and classified in the book \cite{Da1}. This
question goes beyond the scope of the present paper; we hope to
return to it is a later publication.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Acknowledgements}
This work was supported by the Czech Academy of Sciences
within the project K1010104 and the French-Czech AS-CNRS
bilateral program, and by the Ministry of Education of the
Czech Republic within the Japanese-Czech project ME482.
The authors express their gratitude for the hospitality
extended to them and financial support, V.Z. and H.N. in
Department of Theoretical Physics, NPI AS, in \v{R}e\v{z},
and P.E. and H.N. at the Universit\'{e} de la Mediterran\'ee
and in Centre de Physique Th\'eorique, CNRS, in Marseille.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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