Re}\, \langle f,S(\zeta,\tau)f\rangle \geq 0$ for every $f \in
{\cal H}$. For definiteness we use here and in the following the physicist convention about the inner product supposing that it is antilinear in the first argument. For a non-zero $\zeta \in
{\mathbb{C}}$ with $\hbox{\rm Re}\, \zeta \geq 0$, we put also $$\label{Htau}
 H(\zeta) := \zeta^{-1}[I- e^{-\zeta H}]\,.$$ Each element $v(\cdot)$ in $L^2_\mathrm{loc}([0,\infty); {\cal
H})$ is an equivalence class such that any two representatives of it are equal a.e. on $[0, \infty)$. However, at some places we will not avoid an abuse of notation using for a particular representative of such an element the same symbol $v(\cdot)$. At the same time, in the following the convergence of a family of vectors $v(\cdot,\tau)$ to $v(\cdot)$ in the topology of the space $L^2_\mathrm{loc}([0,\infty); {\cal H}) =
L^2_\mathrm{loc}([0,\infty))\otimes {\cal H}$ as $\tau$ tends to zero will be often written as $v(t,\tau) \longrightarrow v(t)$; this will be the case when writing $v(\cdot,\tau) \longrightarrow
v(\cdot)$ would require to introduce a separate symbol for this $v(t,\tau)$ the meaning of which is clear from the context.

The key ingredient of the proof is the following lemma.

\[L1\] $\:(I + S(it ,\tau))^{-1}$ converges to $(I + itH_P)^{-1}P$ as $\tau 