spacetime with a singular brane on the boundary. The $AdS$/CFT setting generates non-trivial constraints. For RS1 and RS2 it generates $1d$ box($0\leq \phi \leq \pi$) and half-line($0 \leq y < \infty$) constraints respectively. These constraints makes the fixed-energy amplitude for $\hat{H}_{RS}$ to be crucially dependent on the boundary conditions(BCs). The combination of these constraints with a singular $\delta$-function potential makes the situation to be complicated. The fascinating fact is that Dirichlet BC requires a coupling constant renormalization to lead a non-trivial fixed-energy amplitude although our case is one-dimensional singular quantum mechanics.

The paper is organized as follows. In section 2 we will consider the free particle case with an half-line constraint and $\delta$-function potential at the boundary as a toy model of RS2 case. In this section we will show how BCs play important roles in this simple singular quantum mechanics. Also we will show why coupling constant renormalization is necessary to lead a non-trivial modification in the fixed-energy amplitude at Dirichlet BC. In section 3 we will compute the fixed-energy amplitude for RS2[@park01-3] which depends on a free parameter $\xi$ where $\xi=0$ and $\xi=1$ correspond respectively to pure Dirichlet and pure Neumann BC cases. We will derive in this 