-
\frac{\hat{G}_{F, b}[a, 0; E] \hat{G}_{F, a}[0, b; E]}{\hat{G}_{F, ab}[0^+, 0; E]}.\end{aligned}$$ where the superscripts $D$ and $N$ stand for Dirichlet and Neumann respectively. The explicit calculation shows $$\begin{aligned}
\label{toy1dn}
\hat{G}_F^D[a, b; E]&=&\frac{e^{-\sqrt{2E} |x - y|}}{\sqrt{2E}} - 
\frac{e^{-\sqrt{2E}(|a| + |b|)}}{\sqrt{2E}}   \\   \nonumber
\hat{G}_F^N[a, b; E]&=&\frac{e^{-\sqrt{2E} |x - y|}}{\sqrt{2E}} + 
\frac{\epsilon(a) \epsilon(b) e^{-\sqrt{2E}(|a| + |b|)}}{\sqrt{2E}}.\end{aligned}$$ One can show easily $\hat{G}_F^D$ and $\hat{G}_F^N$ satisfy the following BCs; $$\begin{aligned}
\label{toy1bcs}
\hat{G}_F^D[a, 0; E]&=&\hat{G}_F^D[0, b; E] = 0
                                   \\   \nonumber
\hat{G}_{F, b}^N[a, 0; E]&=&\hat{G}_{F, a}[0, b; E] = 0.\end{aligned}$$ Then, the general fixed-energy amplitude for $\hat{H}_0^>$ can be obtained by linearly combining $\hat{G}_F^D$ and $\hat{G}_F^N$; $$\label{toy1gen1}
\hat{G}_F^{\xi}[a, b; E] = \xi \hat{G}_F^N[a, b; E] + (1 - \xi)
\hat{G}_F^D[a, b; E]$$ where $\xi(0 \leq \xi \leq 1)$ is real parameter parametrizing the BCs at the origin. Of course $\xi=0$ and $\xi=1$ represent the pure Dirichlet and pure Neumann BCs respectively. Another interesting case is $\xi= 1/2$, in which the contribution of Neumann and Dirichlet have an equal weighting factors. Since the fixed-energy amplitude $\hat{G}_F^{\xi}$ is in general expressed in terms of eigenvalues $E_n$ and eigenfunctions $\phi_n$ of $\hat{H}_0^>$ as follows $$\label{toy1eigen}
\hat{G}_F^{\xi}[a, b; E] = \sum_n \frac{\phi_n(a) \phi_n^{\ast}(b)}{E - E_n},$$ the $\xi=1/2$ case should 