particular, along the sequence $\{\tau_n\}$ with $\tau_n :=
1/m_n$, and since $L^2$ convergence implies pointwise convergence a.e. along a subsequence, there exist a subset $M_f$ of Lebesgue measure zero of the variable $t$ in $[0,\infty)$ and a subsequence $\{\tau_{f,n}\}$ of $\{\tau_n\}$, both dependent on $f$, such that $$(I+ S(it,\tau_{f,n}))^{-1}f \longrightarrow
          (I+ itH_P)^{-1}Pf$$ holds strongly in ${\cal H}$ for $t \in [0,\infty) \setminus M_f$. Since ${\cal H}$ is separable by assumption, we can choose a countable dense subset ${\cal D} = \{f_{\ell}\}_{\ell
=1}^{\infty}$ in ${\cal H}$. Then we infer that for $f_1 \in {\cal
D}$ there exist a set $M_1 := M_{f_1}$ of Lebesgue measure zero and a subsequence $\{\tau_{1,n}\}$ of $\{\tau_n\}$ along which $(I+ S(it,\tau_{1,n}))^{-1}f$ converges to $(I+ itH_P)^{-1}Pf$ for every $t \notin M_1$. Next, for $f_2 \in {\cal D}$ there exist a set $M_2 := M_{f_2}$ of Lebesgue measure zero and a subsequence $\{\tau_{2,n}\}$ of $\{\tau_{1,n}\}$ along which $(I+
S(it,\tau_{2,n}))^{-1}f$ converges to $(I+ itH_P)^{-1}Pf$ for every $t \notin M_2$. Proceeding in this way, we associate in the $\ell$-th step with $f_{\ell} \in {\cal D}$ a set $M_{\ell} :=
M_{f_{\ell}}$ of Lebesgue measure zero and a subsequence $\{\tau_{\ell,n}\}$ of $\{\tau_{\ell -1,n}\}$ along which $(I+
S(it,\tau_{\ell,n}))^{-1}f$ converges to $(I+ itH_P)^{-1}Pf$ for 