calculated from $\hat{G}_{>0}[a, b; E]$ introducing $\delta$- and $\delta^{\prime}$-functions as we did in the previous section; $$\begin{aligned}
\label{rs2gdgn}
\hat{G}_0^D[a, b; E]&=&\hat{G}_{>0}[a, b; E] - 
\frac{\hat{G}_{>0}[a, c; E] \hat{G}_{>0}[c, b; E]}
     {\hat{G}_{>0}[c^+, c; E]}          \\    \nonumber
\hat{G}_0^N[a, b; E]&=&\hat{G}_{>0}[a, b; E] -
\frac{\hat{G}_{>0, b}[a, c; E] \hat{G}_{>0, a}[c, b; E]}
     {\hat{G}_{>0, ab}[c^+, c; E]}.\end{aligned}$$

Inserting Eq.(\[rs2schul\]) into Eq.(\[rs2gdgn\]) one can derive the explicit forms of $\hat{G}_0^D$ and $\hat{G}_0^N$; $$\begin{aligned}
\label{rs2gdgn2}
\hat{G}_0^D[a, b; E]&=&\hat{G}_{>0}[a, b; E] - 2 \sqrt{a b}
\frac{I_{\gamma}(\sqrt{2E} c)}{K_{\gamma}(\sqrt{2E} c)}
K_{\gamma}(\sqrt{2E} a) K_{\gamma}(\sqrt{2E} b)
                                               \\   \nonumber
\hat{G}_0^N[a, b; E]&=&\hat{G}_{>0}[a, b; E] + 2 \sqrt{a b}
\frac{f_I(c, E)}{f_K(c, E)} 
K_{\gamma}(\sqrt{2E} a) K_{\gamma}(\sqrt{2E} b)\end{aligned}$$ where $$\begin{aligned}
\label{fkfi}
f_K(x, E)&=& \frac{\gamma - \frac{1}{2}}{\sqrt{2E} x} K_{\gamma}(\sqrt{2E} x)
+K_{\gamma - 1}(\sqrt{2E} x)    \\   \nonumber
f_I(x, E)&=&I_{\gamma - 1}(\sqrt{2E} x)
- \frac{\gamma - \frac{1}{2}}{\sqrt{2E} x} I_{\gamma}(\sqrt{2E} x).\end{aligned}$$ The useful relation which will be used frequently is $$\label{frequent}
f_K(x, E) I_{\gamma}(\sqrt{2E} x) + f_I(x, E) K_{\gamma}(\sqrt{2E} x)
= \frac{1}{\sqrt{2E} x}.$$

Following Schulman procedure it is straightforward to derive a fixed-energy amplitude for $\hat{H}_{RS2}$; $$\label{rs2schulman}
\hat{G}_{RS2}[a, b; E] = \hat{G}_0^{\xi}[a, b; E]
+ \frac{\hat{G}_0^{\xi}[a, c; E]\hat{G}_0^{\xi}[c, b; E]}
       {\frac{1}{v} - \hat{G}_0^{\xi}[c^+, c; E]}$$ where $\hat{G}_0^{\xi}$ is given in Eq.(\[rs2onepara\]). The most convenient form of $\hat{G}_{RS2}$ is $$\begin{aligned}
