of writing down the general expansion we restrict ourselves here to the special case that there are no bound states, there are two resonances at $z_{R_1}$ and $z_{R_2}$ and there is one second order pole at $z_d=E_d-i\gamma_d$. Then the following basis system expansion holds for $\phi_+\in\Phi_-$: $$\begin{aligned}
\phi^+&=&
|z_d^-\rangle^{(0)}(-2\pi ia_{-2})^{(1)}\langle^+z_d|\phi^+\rangle-
|z_d^-\rangle^{(1)}(-2\pi ia_{-2})^{(0)}\langle^+z_d|\phi^+\rangle
\nonumber\\
&+&\sum_{i=1}^{2}
|z_{R_i}^-\rangle(-2\pi ia_{-1}^{(i)})\langle^+z_{R_i}|\phi^+\rangle
+\int_0^{-\infty_{II}}d\omega|\omega^+\rangle\langle^+\omega|\phi^+\rangle
\label{G22}\end{aligned}$$ the $a_{-2}$, $a_{-1}^{(i)}$ are the expansion coefficients in the Laurent series expansion of the $S$-matrix at the poles $z_d$, $z_{R_1}$ and $z_{R_2}$, respectively.

The important distinction to (\[G16\]) is that this basis system contains Jordan vectors and the Hamiltonian is not diagonal but can only attain the Jordan normal form: $$\left(
\begin{array}{c}
\langle\psi^-|H^\times|z_d^-\rangle^{(0)}\\
\langle\psi^-|H^\times|z_d^-\rangle^{(1)}\\
\langle\psi^-|H^\times|z_{R_1}^-\rangle\\
\langle\psi^-|H^\times|z_{R_2}^-\rangle\\
\langle\psi^-|H^\times|\omega^+\rangle
\end{array}
\right)
=
\left(
\begin{array}{ccccc}
z_d&0\\
1&z_d\\
&&z_{R_1}\\
&&&z_{R_2}\\
&&&&(\omega)\\
\end{array}
\right)
\left(
\begin{array}{c}
\langle\psi^-|z_d^-\rangle^{(0)}\\
\langle\psi^-|z_d^-\rangle^{(1)}\\
\langle\psi^-|z_{R_1}^-\rangle\\
\langle\psi^-|z_{R_2}^-\rangle\\
\langle\psi^-|\omega^+\rangle
\end{array}
\right)
\label{G23}$$ The time evolution of the basis vectors on the r.h.s. of (\[G23\]) is again given by the semigroup (\[G8\]), i.e., they have an arrow of time. However, now in addition to the exponential dependence the time evolution operator also transforms according to (\[ar22-1\]) inside the two dimensional eigenspace ${\cal M}_{z_d}$ with an additional linear time dependence. That second order poles of the $S$-matrix 