fixed-energy amplitude $\hat{G}[x_1, x_2; E]$ by a relation $$\label{example5}
\hat{K}[x_1, x_2; E] = -i \hat{G}[x_1, x_2; -E]$$ where $-E$ in $\hat{G}$ is a usual Euclidean nature. Of course, one can compute the Feynman propagator by taking an inverse Laplace transform to $\hat{G}[x_1, x_2; E]$ and using a relation (\[kerneldef\]).

Extension of Schulman’s procedure to higher dimensional cases is not straightforward due to the infinity arising from origin. In these cases we have to modify Eq.(\[schulman\]) appropriately to escape the ultraviolet divergence[@park95]. Especially, $2d$ case is very interesting because a lot of non-trivial effects are involved in $2d$ $\delta$-function potential such as scale anomaly and dimensional transmutation. In Ref.[@jack91] Jackiw explored the $2d$ $\delta$-function potential system by making use of the physically-oriented coupling constant renormalization and the mathematically-oriented self-adjoint extension[@capri85; @albev88]. He also derived the relation of the renormalized coupling constant to the self-adjoint extension parameter. His result is generalized within a path-integral or Green’s function formalism in Ref.[@park95; @gros93; @gros95; @park96; @park98].

The purpose of this paper is to examine the property of the localized gravity in RS1 and RS2 scenario by treating Eq.(\[grafluc\]) as a Schrödinger equation. In this paper we adopt $AdS$/CFT setting, [*i.e.*]{} single copy of $AdS_5$ 