  We treat a quantum walk (QW) on the line whose quantum coin at each vertex
tends to be the identity as the distance goes to infinity. We obtain a limit
theorem that this QW exhibits localization with not an exponential but a
"power-law" decay around the origin and a "strongly" ballistic spreading called
bottom localization in this paper. This limit theorem implies the weak
convergence with linear scaling whose density has two delta measures at $x=0$
(the origin) and $x=1$ (the bottom) without continuous parts.
