${H}^\times$ (i.e., the extension of the self-adjoint operator ${H}^\dagger$ to $\Phi^\times$) is not diagonalizable, but can only be brought into the normal form of a Jordan block: $$H^\times_{{\cal N}}\Longleftrightarrow\left(
\begin{array}{cccccc}
              z_{{\cal N}}&0&\cdots&\cdots&\cdots&0\\
              1&z_{{\cal N}}&&&&\vdots\\
              0&1&\ddots&&&\vdots \\
              \,\vdots&&\ddots&\ddots&&\vdots\\
              \,\vdots&&&\ddots&\ddots&0\\
              0&0&\cdots&\cdots&1&z_{{\cal N}}\end{array}
\right)$$

From this follows that the matrix representation of the time evolution operator (\[G8\]) on the ${\cal N}$-dimensional eigenspace ${\cal M}_{z_{\cal N}}$ is given by $$\begin{minipage}{0.9\textwidth}
\epsfxsize=\textwidth
\epsffile{wzor.eps}
\end{minipage}
\label{ar22-1}$$ which can also be written as $$\begin{aligned}
&&\mbox{e}^{-itH^\times}\mid z^-_{{\cal N}}\rangle^{(k)}=
\mbox{e}^{-itz_{{\cal N}}}\left\{\mid z^+_{{\cal N}}\rangle^{(k)}
+\frac{(-it)}{1!}\mid z^+_{{\cal N}}\rangle^{(k-1)}+\cdots\right.
\nonumber\\
&&+\left.
\vphantom{\mbox{e}^{-itH^\times}\mid z^-_{{\cal N}}\rangle^{(k)}=
\mbox{e}^{-tz_{{\cal N}}}\left\{\mid z^+_{{\cal N}}\rangle^{(k)}
+\frac{(-it)}{1!}\mid z^+_{{\cal N}}\rangle^{(k-1)}+\cdots\right.
}
\frac{(-it)^l}{l!}\mid z^+_{{\cal N}}\rangle^{(k-l)}+
\cdots
+\frac{(-it)^k}{k!}\mid z^+_{{\cal N}}\rangle^{(0)}
\right\}
\,\,\,\,\mbox{for $t\geq0$ only}.
\label{ar22-2}\end{aligned}$$ This means that whereas the zeroth order Gamow state only decays exponentially with time, the $k$-th order Gamow vector $\mid z^-_{{\cal N}}\rangle^{(k)}$ evolves into a superposition of lower order Gamow vectors. After a long time (relative to the scale set by $\hbar/\Gamma$) the most significant term is the zeroth order Gamow vector $\mid z_{{\cal N}}^-\rangle^{(0)}$, whose time dependence is given by $\mbox{e}^{-itz_{{\cal N}}}t^{k}$.

The complex basis vector expansion and some of its consequences in physical applications
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The most important result of the new mathematical theory of quantum 