the eigenvalue equation $$\int d^4K\Delta ^{-1}_{ab}(P,K;q)\Phi _{b}^{\nu }(K,q)=\alpha (q^2)\Phi
_{a}^{\nu }(P,q)\label{2.18}$$ to be obtained. The quantity $\Delta ^{-1}_{ab}(P,K;q)$ is defined as $$\Delta ^{-1}_{ab}(P,K;q)\equiv \frac{9}{2}\int \frac{d^4r}{(2\pi
)^4}\frac{e^{-i(P-K)r}}{g^2D(r)}\delta _{ab}
+\delta ^{(4)}(P-K)\mbox{tr}\left[ M^aG(K_{+})M^bG(K_{-})\right] \label{2.19}$$ and plays the role of an inverse propagator for the $q\bar{q}$ composite system, as has been demonstrated in previous work on the hadronization of quark field-theory models[@localization]. That description is extended here to include the more general structure in (\[2.17\]). The eigenfunctions are defined in terms of the coefficient functions, $\phi ^{(i)}$, as $\Phi _{1}^{\nu }\equiv \phi
^{(1)}_{\nu }$, $\Phi _{2-5}^{\nu }\equiv \phi ^{(2)}_{\alpha \nu }$, and $\Phi _{6-9}^{\nu }\equiv \phi ^{(3)}_{\alpha \nu }$. The matrices $M^a\equiv {\bf 1}_C{\bf 1}_F\left\{ {\bf 1},i\gamma _{\mu
}/\sqrt{2},i\gamma _{5}\gamma _{\mu }/\sqrt{2}\right\} $ include unity in the color and two-component flavor space, and the trace is taken to include their indices as well as those of the Dirac matrices. The homogeneous equation (\[BS\]) is regained through the requirement that the eigenvalue vanishes on the mass shell, that is, $\alpha
(q^2=-M_V^2)=0$. The normalization condition for the eigenfunctions is obtained by multiplying (\[2.17\]) on the left by $\Phi _{a}^{\mu
}(P,-q)$ and integrating over relative momentum $P$, which gives $$\begin{aligned}
\int d^4Pd^4K\Phi _{a}^{\mu }(P,-q)\Delta ^{-1}_{ab}(P,K;q)\Phi
_{b}^{\nu }(K,q)&=&\alpha 