$u \in D[HP]$, while $H^{1/2}Pu_0 = H^{1/2}u_0 \not= 0$ by assumption.

To describe our results, we denote by $L^2_\mathrm{loc}([0,\infty); {\cal H}) =
L^2_\mathrm{loc}([0,\infty)) \otimes {\cal H}$ the Fréchet space of the ${\cal H}$-valued strongly measurable functions $v(\cdot)$ on $[0,\infty)$ such that $\|v(\cdot)\|$ is locally square integrable there, with the topology induced by the semi-norms $v\mapsto \bigl(\int_0^{T_{\ell}} \|v(t)\|^2 dt\bigr)^{1/2}$ for a countable set $\{T_{\ell}\}_{\ell =1}^{\infty}$ of increasing positive numbers accumulating at infinity, $\lim_{\ell \rightarrow
\infty} T_{\ell} = \infty$. In a similar way one defines the Fréchet space $L^2_\mathrm{loc}({\mathbb{R}}; {\cal H}) =
L^2_\mathrm{loc}({\mathbb{R}}) \otimes {\cal H}$.

Our main result can be stated as follows:

\[main\] Let $H$ be a nonnegative self-adjoint operator on a separable Hilbert space ${{\mathcal H}}$ and $P$ an orthogonal projection. Let $t
\mapsto P(t)$ be a strongly continuous function whose values are orthogonal projections in ${\cal H}$, defined in some neighborhood of zero, with $P(0) =: P$. Moreover, suppose that $D[H^{1/2}P(t)]
\supset D[H^{1/2}P]$ and $\lim_{t \rightarrow 0} \|H^{1/2}P(t)v\|
= \|H^{1/2}Pv\|$ holds for $v \in D[H^{1/2}P]$. If the operator $H_P$ specified above is densely defined in the whole Hilbert space ${{\mathcal H}}$, then for every $f \in {\cal H}$ and $\varepsilon=\pm
1$ it holds that $$\begin{aligned}
&&[P(1/n)\exp(-i\varepsilon tH/n) P(1/n)]^n 