greater variety of solutions; (this goes back to Dirac (kets $\mid E \rangle$), Gamow (exponentially decaying “state" vectors $\mid E - i \Gamma/2 \rangle$) and Peierls (purely outgoing boundary conditions). These new vectors are in the rigged Hilbert space, $\mid E
\rangle, \mid E - i \Gamma/2 \rangle\, \in \Phi^{\times} \supset {\cal H}
\supset \Phi$, but not in the Hilbert space $\cal H$. Distinct initial-boundary conditions for state vectors (e.g., [*in*]{}-states $\phi^{+}$ of a scattering experiment) and observables $| \psi^-\rangle
{\langle}\psi^- |$ (e.g., so-called [*out*]{}-states $\psi^-$ of a scattering experiment) lead to two different rigged Hilbert spaces [@R1], whose precise mathematical properties had been defined earlier [@R24]: $$\begin{aligned}
&{\Phi}_-\subset{{\cal H}}\subset{\Phi}^\times_-,\qquad\qquad&
\mbox{for ensembles or states}\label{G4a}\\
&{\Phi}_+\subset{{\cal H}}\subset{\Phi}^\times_+,\qquad\qquad&
\mbox{for observables or effects}
\label{G4b}\end{aligned}$$ The Hilbert space ${\cal H}$ is the same in both RHS’s (\[G4a\]) and (\[G4b\]) and $\Phi$ of (\[3a\]) is $\Phi=\Phi_-+\Phi_+$ with $\Phi_-\cap\Phi_+\neq\emptyset$.

In (\[G4a\]), $\Phi_-$ describes the possible state vectors (preparation apparatus, e.g., $\phi^{\mbox{\rm\scriptsize in}}$ or $\phi^{+}$ of a scattering experiment) and $\Phi_{+}$ in (\[G4b\]) describes the possible observables (e.g., $| \psi^{\mbox{\rm\scriptsize out}}\rangle
{\langle}\psi^{\mbox{\rm\scriptsize out}} |$ or $|\psi^-\rangle{\langle}\psi^- |$ of a scattering experiment).

For the typical scattering experiment the physical meaning of $\Phi_-\ni\phi^+$ is depicted in Fig. 1. The 