definition of Dirac kets was given in 1966 [@R9], the Gamow vectors were introduced about 1976 [@R10],[@R11]; a generalization of Gamow vectors to higher order poles of the $S$-matrix was given in 1995 [@R12]. An ${\cal
N}$-th order $S$-matrix pole at the complex energy $z_{{\cal N}}=E_{{\cal N}}-i\gamma_{{\cal N}}$ has ${{\cal N}}$ Gamow vectors of order $0,1,\ldots k\ldots ({{\cal N}}-1)$: $$|z_{{\cal N}}^-\rangle^{(0)},\,\,
|z_{{\cal N}}^-\rangle^{(1)},{\ldots,}|z_{{\cal N}}^-\rangle^{(k)},
{\ldots,}|z_{{\cal N}}^-\rangle^{({{\cal N}}-1)}
\label{G11}$$ associated with it. The $k$-th order Gamow vector $|z_{{\cal N}}^-\rangle^{(k)}$ is a Jordan vector of degree $(k+1)$, i.e. it fulfills the eigenvalue equations [@R59] $$\begin{aligned}
&&({H}^\times-z)^k|z_{{\cal N}}^-\rangle^{(k)}=0;\nonumber\\
&&{H}^\times|z_{{\cal N}}^-\rangle^{(k)}=z_{{\cal N}}|z_{{\cal N}}^-\rangle^{(k)}
+|z_{{\cal N}}^-\rangle^{(k-1)}\,\,\,\,\mbox{for}\,\,\,\,
k=0,1,\ldots,({{\cal N}}-1)\end{aligned}$$ These equations are, like the eigenvector equation for Dirac kets and for Gamow vectors (= Gamow vectors of order 0 = Jordan vectors of degree 1), understood as generalized eigenvector equations (i.e., functionals) over the space $\Phi_+$: $$\begin{aligned}
{\langle}{H}\psi^-|z^-_{{{\cal N}}}\rangle^{(k)}
&\equiv&{\langle}\psi^-|{H}^\times|z^-_{{{\cal N}}}\rangle^{(k)}\nonumber\\
&=&z_{{{\cal N}}}{\langle}\psi^-|z^-_{{{\cal N}}}\rangle^{(k)}
+\langle\psi^-\mid z^-_{{{\cal N}}}\rangle^{(k-1)}
\,\,\,\,\mbox{for all}\,\,\psi_-\in\Phi_+.
\label{G12}\end{aligned}$$ This means $|z_{{\cal N}}^-\rangle^{(k)}\in\Phi_+^\times$, and the ${{\cal N}}$-th order $S$-matrix pole is associated to a ${{\cal N}}$-dimensional subspace ${\cal M}_{z_{{{\cal N}}}}\subset\Phi_+^\times$, spanned by the $|z_{{\cal N}}^-\rangle^{(k)}$, $k=0,1,\ldots,({{\cal N}}-1)$, i.e., to the set of all $$|z_{{\cal N}}^-\}=\sum_{k=0}^{{{\cal N}}-1}|z_{{\cal N}}^-\rangle^{(k)}c_k,
\,\,\,\,c_k\in{\cal C}.$$ On ${\cal M}_{z_{{\cal N}}}\subset\Phi_+^\times$ the Hamiltonian 