  From the viewpoint of the singular quantum mechanics the effect of the
energy-dependent coupling constant for $\delta$-function potential is examined.
The energy-dependence of the coupling constant naturally generates the
time-derivative in the boundary condition of the Euclidean propagator. This is
explicitly confirmed by making use of the simple 1d model. The result is
applied to the linearized gravity fluctuation equation for the brane-world
scenario with 4d induced gravity. Our approach generates $5d$ Newton potential
at a certain intermediate range of distance between two test massive sources.
For other range of distance 4d Newton potential is recovered.
