\xi_1) \bigg\{\cosh \sqrt{2E} (L - |a - b|) - \cosh 
                                         \sqrt{2E} ((a + b) - L) \bigg\}
                                                  \Bigg]
                                                     \\  \nonumber
\hat{G}_0^{\xi_1, N}[a, b; E]&=&
\frac{1}
     {\sqrt{2E} [(1- \xi_1)\cosh \sqrt{2E} L + xi_1 \sinh \sqrt{2E} L]}
                                                          \\   \nonumber
& & \times
\Bigg[ \xi_1 \bigg\{\cosh \sqrt{2E} (L - |a - b|) + \cosh \sqrt{2E} ((a + b)
                                                              - L) \bigg\}
                                                    \\  \nonumber
& & \hspace{1.0cm}
+ (1 - \xi_1) \bigg\{\sinh \sqrt{2E} (L - |a - b|) + \sinh 
                                            \sqrt{2E} ((a + b) - L) \bigg\}
                                                    \Bigg].\end{aligned}$$ Inserting eq.(\[toy2explicit\]) into Eq.(\[toy2box1\]) we get $$\begin{aligned}
\label{toy2box2}
\hat{G}_0^{Box}[a, b; E]&=& \frac{1}{\sqrt{2E}}
\Bigg[ \left( \mu(\xi_1, \xi_2) + \mu(1 - \xi_1, 1 - \xi_2) \right)
                               \cosh \sqrt{2E} (L - |a - b|)
                                              \\   \nonumber
& & \hspace{1.0cm} 
+ \left( \mu(\xi_1, \xi_2) - \mu(1 - \xi_1, 1 - \xi_2) \right)
                               \cosh \sqrt{2E} ((a + b) - L)
                                               \\   \nonumber
& & \hspace{1.0cm} 
+ \left( \nu(\xi_1, \xi_2) + \nu(1 - \xi_1, 1 - \xi_2) \right)
                               \sinh \sqrt{2E} (L - |a - b|)
                                               \\   \nonumber
& & \hspace{1.0cm} 
+ \left( \nu(\xi_1, \xi_2) - \nu(1 - \xi_1, 1 - \xi_2) \right)
                               \sinh \sqrt{2E} ((a + b) - L)
                                                           \Bigg]\end{aligned}$$ where $$\begin{aligned}
\label{toy2munu}
\mu(z, w)&=& \frac{z w}{(1 - z) \cosh \sqrt{2E} L 