called $\phi_{\mbox{\rm\scriptsize bg}}^+$ and which is not taken into consideration in any of the finite dimensional effective theories of complex Hamiltonians, in particular not in the Lee, Oehme, Yang theory [@R13] of the neutral Kaon system. This term, which comes from the integral along the negative real axis in the second sheet of the $S$-matrix, can be shown to be also decaying, i.e., $|{\langle}\psi^-|\phi_{\mbox{\rm\scriptsize bg}}^+(t)\rangle|\rightarrow0$ for $t\rightarrow\infty$ for every $\psi^-\in\Phi_+$, but it decays more slowly than the exponential [@R20]. Thus if one takes for $\phi^+$ the $|K^0\rangle$ state prepared e.g., by the reaction $p\pi^-\rightarrow\Lambda K^0$ and for the observables $|\psi^-\rangle{\langle}\psi^-|$ projectors on the $\pi^+\pi^-$ space one obtains according to the [*exact*]{} equation (\[G19\]) also a term $|{\langle}\pi^+\pi^-|\phi_{\mbox{\rm\scriptsize bg}}^+(t)\rangle|$. This term vanishes more slowly than to the rapidly disappearing $\mbox{e}^{-i\Gamma_St}|{\langle}\pi^+\pi^-|K^0_1\rangle|$. This may provide an alternative mechanism to explain the $\pi\pi$ decay mode of the prepared $K^0$ long after the $K_S^0=K_1^0$ has vanished.

The [*third*]{} step outside the space to obtain the basis of eigenvectors is the [*general complex basis vector*]{} expansion. It includes in addition to the ordinary Gamow kets (\[G9\]) also higher order Gamow kets (\[G12\]) which occur when (and if) the $S$-matrix has poles of order ${\cal N}>1$. Instead 