$\phi^+$), and upon some phase convention. All the vectors in the generalized basis system expansion are (generalized) eigenvectors of the exact Hamiltonian, and, in particular, the Gamow vectors $\psi^G_{L,S}$ are eigenvectors of the exact Hamiltonian $H$, with complex eigenvalue $\left( E_L -i\,{{\Gamma}_L / 2} \right)$ and $\left( E_S-i\,{{\Gamma}_S / 2}\right)$, respectively.

We now apply the time evolution operator to equation (\[N25\]). Since the $\psi_i^G$ are elements of $\Phi^\times_+$ we can only apply the operator $U^\times_+(t)$ of (\[G8\]) to it and we obtain: $$\phi^+ (t)\equiv e^{-iH^\times t}\phi^+=
   e^{-i(E_L-i\,{\Gamma}_L /2)t}\,\psi^G_Lb_L\; +\;
e^{-i(E_S-i\,{\Gamma}_S /2)t}\,\psi^G_Sb_s\; +\;\phi^+_{\mbox{\rm\scriptsize bg}}
(t)\; ; t\ge 0
\label{G19}$$ Since the time evolution semigroup (\[G8\]) has the restriction $t\ge
0$, the same restriction must be used for (\[G19\]). $\phi^+_{\mbox{\rm\scriptsize bg}} (t)$ is the time evolved background term $$\phi^+_{\mbox{\rm\scriptsize bg}} (t)\equiv
\int^{-\infty_{II}}_{0} dE\; e^{-iEt}\,
|E^+\rangle{\langle}^+E|\phi^+\rangle \;.$$ These equations are understood as a functional equation over all $\psi^-\in{\Phi}_+$. This means that $\phi^+ (t)\in\Phi_-\subset\Phi_+^\times $ can be used to obtain ${\langle}\psi^-|\phi^+ (t)\rangle$, whose modulus square is the probability to find the time evolved state by a detector that detects the observable $|\psi^-\rangle{\langle}\psi^-|$ for any $\psi^-\in{\Phi}_+$, but [*not*]{} for a $\psi^-\in{\Phi}_-$.

The result (\[G19\]) means that the time evolution of a superposition of two (or 