eigenvectors of the exact Hamiltonian $H=K+V$, (describing the bound states) $H|E_n)=E_n|E_n)$, and $|E^+\rangle$ are the generalized eigenvectors (Dirac kets) of $H$ corresponding to the continuous spectrum (describing scattering states). The integration extends over the spectrum of $H$: $0\le E <\infty$; and in place of the $|E^+\rangle$ one could also have chosen the $|E^-\rangle$, if the [*out*]{}-wavefunctions are more readily available.

The [*second*]{} step is the “complex basis vector expansion” For every $\phi^+\in\Phi_-$ (a similar expansion holds also for every $\psi^-\in\Phi_+$) one obtains for the case of a finite number of resonances poles at the positions $z_{R_i}\,,\;i=1,\;2,\cdots N$, the following basis system expansion: $$\begin{aligned}
\phi^+&=&
\int_0^{-\infty_{II}}d{\omega}\;|{\omega}^+\rangle{\langle}^+{\omega}|\phi^+\rangle\;+\;
\sum_{i=1}^N\,|z_{R_i}^-\rangle\;2\pi{\Gamma}_i \;{\langle}^+z_{R_i}|\phi^+\rangle\nonumber\\
&+&\sum_n |E_n)(E_n|\phi)\,\,\,\,
\mbox{for}\,\,\,\, \phi^+\in{\Phi}_-
\label{G16}\end{aligned}$$ where $|z_{R_i}^-\rangle\sqrt{2\pi\Gamma_i}=\psi^{G_i}\in\Phi^\times_+$ are Gamow kets (\[G9\]) representing decaying states (\[G10\]).

If we assume that there are two decaying states $R_1 =S$ and $R_2 =L$ and no bound states, then the pure state (prepared by the experimental apparatus) has according to (\[G16\]) the following representation in terms of the Gamow vectors $\psi_L^G=\mid z_L^-\rangle\sqrt{2\pi\Gamma_L}$, $\psi_S^G=\mid z_S^-\rangle\sqrt{2\pi\Gamma_S}$ and the remaining part which we call $\phi^+_{\mbox{\rm\scriptsize bg}}$: $$\phi^+=\psi^G_L\,b_L\;+\;\psi^G_S\,b_S\;+\;\int^{-\infty_{II}}_0 dE\;
|E^+\rangle{\langle}^+E|\phi^+\rangle.
\label{N25}$$ In here $b_L$ and $b_S$ are some complex numbers that depend upon the “normalization” of the Gamow vectors $\psi^G_{L,S}$ (and of 