\label{toy2final}
\hat{G}_{\delta, \xi_1=\xi_2=0}^{Box}[a, b; E]&=&\hat{G}_0^{DD}[a, b; E] 
                                                    \\   \nonumber
& &
+ 2 (v_1^{ren} + \sqrt{2E} \coth \sqrt{2E}L)^{-1}
\frac{\sinh \sqrt{2E} (L - a) \sinh \sqrt{2E} (L - b)}{\sinh^2 \sqrt{2E} L}
                                                     \\    \nonumber
&-&
2 \left[ (v_1^{ren} + \sqrt{2E} \coth \sqrt{2E} L) (v_2^{ren} - 
                                                \sqrt{2E} \coth \sqrt{2E} L)
       + \frac{2E}{\sinh^2 \sqrt{2E} L}       \right]^{-1}
                                                      \\    \nonumber
&\times&
\Bigg[(v_1^{ren} + \sqrt{2E} \coth \sqrt{2E} L)
      \frac{\sinh \sqrt{2E} a \sinh \sqrt{2E} b}{\sinh^2 \sqrt{2E} L}
                                                       \\   \nonumber
& &   + \sqrt{2E}
      \frac{\sinh \sqrt{2E} a \sinh \sqrt{2E} (L - b) + 
            \sinh \sqrt{2E} (L - a) \sinh \sqrt{2E} b}
           {\sinh^3 \sqrt{2E} L}
                                                        \\   \nonumber
& &   + 2E (v_1^{ren} + \sqrt{2E} \coth \sqrt{2E} L)^{-1}
       \frac{\sinh \sqrt{2E} (L - a) \sinh \sqrt{2E} (L - b)}
            {\sinh^4 \sqrt{2E} L} \Bigg].    \end{aligned}$$ In the next section we will apply the analysis in this toy model to the RS1 scenario.

Fixed-energy Amplitude for RS1
==============================

In this section we will examine the fixed-energy amplitude for RS1 whose linear gravitational fluctuation is given in Eq.(\[grafluc\]) and Eq.(\[potential\]). The Hamiltonian for RS1 can be read from these equations easily $$\begin{aligned}
\label{rs1hamil1}
\hat{H}_{RS1}&=&\hat{H}_0 - v_1 \delta(z) + v_2 \delta(z - z_0)
                                                   \\   \nonumber
\hat{H}_0&=&-\frac{1}{2} \partial_z^2 + 
\frac{g}{(|z| + c)^2}.\end{aligned}$$ Of course, the exact RS1 