energy and width.

    2.  They satisfy the following exponential decay law for $t \geq 0$ only: $$\begin{aligned}
          &W^{G}(t)
          &=
          \mbox{``}\, e^{-iHt/\hbar} \,\mbox{''}
          \left|\psi^{G}\right\rangle
          \left\langle\psi^{G}\right|
          \mbox{``}\, e^{iHt/\hbar} \,\mbox{''}\nonumber\\
          &&=
          e^{-i\left(E_{R} - i\,{\Gamma /2}\right)t/\hbar}
          \left|\psi^{G}\right\rangle
          \left\langle\psi^{G}\right|
          e^{i\left(E_{R} + i\,{\Gamma /2}\right)t/\hbar}\nonumber\\
          &&=
          e^{-\Gamma t/\hbar}\,
          W^{G}(0)
          \label{G10}\end{aligned}$$

    3.  They have a Breit-Wigner energy distribution.

    4.  They obey an exact Golden Rule of which Fermi’s Golden Rule is the Born approximation.

    5.  They are associated with a pole at $z_R$ in the second sheet of the analytically continued $S$-matrix. They are derived as the functionals of the pole term of the $S$-matrix.

In the absence of a vector description of resonances in the Hilbert space formulation, the pole of the $S$-matrix has commonly been taken as the definition of a resonance. Since in the RHS formulation the Gamow vectors are derived from the pole term of the $S$-matrix [@R11], these vectors $\mid z_R^-\rangle\in\Phi_+^\times$ define decaying resonance states as autonomous microphysical entities. (There are also Gamow vectors $|{z^*_R}^+\rangle$, $z^*_R = E_R + i
\Gamma/2$ associated with the pole at $z^*_R$, which have an exponentially growing semi-group evolution for $- \infty <t \leq 0$).

Gamow-Jordan vectors — a mathematical actuality and a physical possibility.
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The mathematical 