Hence, the asymptotic solutions of ${\bm w}^p$ are given by plane waves. We take the two orthogonal solutions ${\bm w}^p_{{\rm C}(\pm )}$ having the asymptotic behavior $$\begin{aligned}
&&i{\bm w}^p_{{\rm C}(\pm )}\rightarrow 
\Bigg\{
\begin{array}{ll}
\rho_{\pm}^pe^{\pm ip\eta_{\rm C}}+e^{\mp ip\eta_{\rm C}}\,,
& (\eta_{\rm C}\rightarrow \pm\infty )\\
\sigma_{\pm}^pe^{\mp ip\eta_{\rm C}}\,,
& (\eta_{\rm C}\rightarrow \mp\infty )\\
\end{array}
\,,
\label{eq:scattering amplitude}\end{aligned}$$ as independent solutions. The reflection and transmission coefficients satisfy the Wronskian relations: $$\begin{aligned}
|\rho_{\pm}^p|^2+|\sigma_{\pm}^p|^2=1\,,\ 
\sigma_{+}^p=\sigma_{-}^p\,,\ 
\sigma_{+}^p\overline{\rho_{-}^p}
+\overline{\sigma_{-}^p}\rho_{+}^p=0\,.
\label{eq:Wronskian}\end{aligned}$$ Using the relation between the coordinates , we analytically continue Eq. (\[eq:w\^p eq\]) to the region R. Then, we obtain the evolution equation for ${\bm w}^p$, $$\begin{aligned}
\Biggl[-\frac{{{\rm d}}^{2}}{{{\rm d}}\eta_{\rm R}^2}
+U_T(\eta_{\rm R})-p^2\Biggr]{\bm w}^p=0\,.
\label{eq:w^p eq in R}\end{aligned}$$ Similarly, the analytic continuations of the two independent modes to the region R are given by $$\begin{aligned}
&&{\bm w}^p_{{\rm R}(+)}
=e^{\pi p/2}\rho_{+}^p\overline{\tilde{\bm w}^p}
+e^{-\pi p/2}\tilde{\bm w}^p
\,,\\
&&{\bm w}^p_{{\rm R}(-)}
=e^{\pi p/2}\sigma_{-}^p\overline{\tilde{\bm w}^p}
\,.\end{aligned}$$ where $\tilde{\bm w}^p$ is the solution of Eq.  that asymptotically behaves like $\tilde{\bm w}^p\rightarrow e^{ip\eta_{\rm R}}$ in the limit $\eta_{\rm R}\rightarrow -\infty$. The amplitude of the fluctuation is proportional to the sum of squares of these two independent modes, which is evaluated as $$\begin{aligned}
&&\bigl|{\bm w}^p_{{\rm R}(+)}\bigl|^{2}
+\bigl|{\bm w}^p_{{\rm R}(-)}\bigl|^{2}
\nonumber\\
&&\ \ 
=2\biggl[\bigl|\tilde{\bm w}^p\bigl|^{2}\cosh\pi 