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$\newcommand{\kk}[1]{|#1\rangle}
\newcommand{\bb}[1]{\langle #1}
\newcommand{\dd}[1]{\delta_{#1}}
\newcommand{\molp}{\Omega^{(+)}}
$
The quantity $T(E\alpha,E'\alpha')\equiv\bb{E\alpha}|V\molp\kk{E'\alpha'}$ occurs very frequently in scattering theory and is called the {\em off-shell} {\bf transition amplitude} or the off-shell T-matrix.
The S-matrix formula identifies T-matrix :
\begin{eqnarray*} \bb{E\alpha}|S\kk{E'\alpha'}=\delta(E-E')\dd{\alpha\alpha'}
-2\pi i\delta(E-E')T(E\alpha,E'\alpha'). \end{eqnarray*}
The first term (matrix element of identity) refers to ``no scattering" because if the final state $E',\alpha'$ is different from initial $E,\alpha$ then the first term is zero. The second term denotes transition probability amplitude. But the transition amplitude here occurs with the energy conserving delta function and is actually the {\em on-shell} transition amplitude $T_E(\alpha,\alpha')\equiv T(E\alpha,E\alpha')$.
