\hspace{2.0cm} \mbox{(RS1)}     \\   \nonumber
h(x, y)&=&e^{-\frac{k}{2} |y|} \hat{\psi} e^{ipx}
\hspace{2.3cm} \mbox{(RS2)}\end{aligned}$$ where $m^2 = -p^2$ and $$\begin{aligned}
\label{changeofv}
z&=&\epsilon(\phi) \frac{e^{k r_c |\phi|} - 1}{k}  
\hspace{2.8cm} \mbox{(RS1)}    \\   \nonumber
z&=&\epsilon(y) \frac{e^{k|y|} - 1}{k}.
\hspace{3.1cm} \mbox{(RS2)}\end{aligned}$$ Since all components are same, Lorentz indicies $\mu$ and $\nu$ are suppressed in Eq.(\[relation1\]).

When deriving the linearized fluctuation equation (\[grafluc\]), we have used the RS gauge choice $$\label{gauge}
h^{\mu}_{\nu,\mu} = h^{\mu}_{\mu} = 0, 
\hspace{2.0cm}
h_{55} = h_{\mu 5} = 0$$ for each RS scenario. This gauge choice, however, generally generates a non-trivial bending effect on the brane[@garr00]. The bending effect usually makes the linearized fluctuation equation (\[grafluc\]) to be non-homogeneous form, [*i.e.*]{} $(\hat{H}_{RS} - m^2/2) \hat{\psi} \neq 0$. Thus, inclusion of the bending effect makes the stroy to be more complicated. In this paper we will not consider the bending effect for simplicity.

The linearized fluctuation equation (\[grafluc\]) looks like usual Schrödinger equation. From the purely mathematical point of view the Hamiltonian operator $\hat{H}_{RS}$ in Eq.(\[grafluc\]) is a singular operator due to the singular $\delta$-function potential in $V_i$. In the path-integral framework[@fey65; @schul81] the $1d$ $\delta$-function potential was treated by Schulman about one and half decades ago as follows[@schul86].

Let us consider $1d$ Hamiltonian $$\label{example1}
