If $c = 0$, the Euclidean propagator $G_{>0}[a, b; t]$ and the corresponding fixed-energy amplitude $\hat{G}_{>0}[a, b; E]$ for Hamiltonian (\[rs2hamil2\]) are given at Ref.[@schul81]; $$\begin{aligned}
\label{rs2schul}
G_{>0}[a, b; t]&=&\frac{\sqrt{a b}}{t} e^{-\frac{a^2 + b^2}{2 t}} 
I_{\gamma}\left( \frac{a b}{t} \right)    \\   \nonumber
\hat{G}_{>0}[a, b; E]&=& 2 \sqrt{a b} 
I_{\gamma}\left( \sqrt{\frac{E}{2}} [(a + b) - |a - b|] \right)
K_{\gamma}\left( \sqrt{\frac{E}{2}} [(a + b) + |a - b|] \right)\end{aligned}$$ where $I_{\gamma}(z)$ and $K_{\gamma}(z)$ are the usual modified Bessel functions, and $\gamma = \sqrt{1 + 8 g} / 2$.

The main problem for the computation of the fixed energy amplitude for $\hat{H}_0$ in Eq.(\[rs2hamil2\]) is how to adopt an asymmetric constraint $x \geq c$ in terms of $x$. However, this is already explained at the previous section by introducing an infinite energy barrier. In the asymmetric barrier the fixed-energy amplitude will be dependent on the BC at $x = c$. Hence, the final form will be one parameter family type $$\label{rs2onepara}
\hat{G}_0^{\xi}[a, b; E] = \xi \hat{G}_0^N[a, b; E] + 
(1 - \xi) \hat{G}_0^D[a, b; E]$$ where $\hat{G}_0^N$ and $\hat{G}_0^D$ are the fixed-energy amplitudes which obey the Neumann and Dirichlet BCs respectively at $x = c$. Of course, $\hat{G}_0^N$ and $\hat{G}_0^D$ can be 