more) Gamow states does not regenerate one Gamow state from the other, or from the background $\phi^+_{\mbox{\rm\scriptsize bg}} (t)$. In particular, if the state $\phi^+$ can be prepared such that at some time $t_0 \ge 0$ the background term $\phi^+_{\mbox{\rm\scriptsize bg}}
(t)$ is practically zero, then it will remain zero for all $t >t_0$, and the two Gamow states will evolve separately with their separate exponential laws without regenerating each other: $$\phi^+ (t)\approx e^{-iE_L t}\,e^{-({\Gamma}_L /2)t}\,\psi^G_L\,b_L \; +\;
e^{-iE_S t}\,e^{-({\Gamma}_S /2)t}\,\psi^G_S\,b_S
\label{G21}$$ Approximations like (\[G21\]) have been used for the time evolution of a two-resonance system (like the $K_L$–$K_S$-system with $\phi^+(t)$ representing the $K^0$ state [@R13]) in theories with “effective Hamiltonians” given by $2\times2$ complex diagonizable matrix. These effective theories are usually legitimized by the Wigner-Weisskopf approximation [@R14]. In our irreversible quantum theory the expression (\[G19\]) is exact and it justifies to some extent the effective theory(\[G21\]). And (\[G19\]) shows that the problem of “deviation from the exponential decay law” or “vacuum regeneration of $K_S$ from $K_L$” [@R19] arises from the artifacts of the Hilbert space mathematics and can be overcome in the exact theory using the rigged Hilbert space.

However there is an extra term in (\[G19\]) which we 