projection. If the operator $H_P$ specified above is densely defined, then for every $f \in {\cal H}$ and $\varepsilon = \pm 1$ we have in the topology of $L_\mathrm{loc}^2({\mathbb{R}}; {\cal H})$ the limiting relation $$[P\exp(-i\varepsilon tH/n) P]^n f \longrightarrow
  \exp(-i\varepsilon tH_P)\,Pf \label{c:product-s}$$ for $\:n\rightarrow \infty$ as well as its nonsymmetric counterparts obtained by setting $P(1/n)=P$ in (\[product-ns\]) and (\[product-ns2\]).

&gt;From the viewpoint of quantum Zeno effect described in the introduction the optimal result would be a strong convergence on $\mathcal{H}$ for a fixed value of the time variable, moreover uniformly on each compact interval in $t$. Our Theorem \[main\] implies the following weaker result on pointwise convergence.

\[cor2\] Under the same hypotheses as in Theorem \[main\], there exist a set $M \subset {\mathbb{R}}$ of Lebesgue measure zero and a strictly increasing sequence $\{n'\}$ of positive integers along which we have $$\begin{aligned}
&& [P(1/n')\exp(-i\varepsilon tH/n') P(1/n')]^{n'} f
 \longrightarrow
 \exp(-i\varepsilon tH_P)\,Pf\,, \label{ae:product-s} \\
&& [P(1/n')\,\exp(-i\varepsilon tH/n') ]^{n'} f
 \longrightarrow
\exp(-i\varepsilon tH_P)\,Pf\,, \label{ae:product-ns}\\
&& [\,\exp(-i\varepsilon tH/n')\,P(1/n')]^{n'} f
 \longrightarrow
\exp(-i\varepsilon tH_P)\,Pf\,, \label{ae:product-ns2}
 \end{aligned}$$ for every $f \in \mathcal{H}$, strongly in ${\cal H}$ for all $t
\in {\mathbb{R}}\setminus M$.

As we have indicated above, one need not resort to subsequences in the particular case when 