Scattering theory between the fractional power $H_0=\kappa^{-1}(-\Delta)^{\kappa/2}$ $(\kappa\ge 1$) of negative Laplacian and the Hamiltonian $H=H_0+V$ perturbed by shortand long-range potentials considered in [14] is revisited and a new proof of the existence and asymptotic completeness of wave operators is given with utilizing the smooth operator technique.