We consider a position-dependent coined quantum walk on $$\mathbb {Z}$$Z and assume that the coin operator C(x) satisfies $$\begin{aligned} \Vert C(x) - C_0 \Vert \le c_1|x|^{-1-\epsilon }, \quad x \in \mathbb {Z}\setminus \{0\} \end{aligned}$$?C(x)-C0?≤c1|x|-1-∈,x?Z\{0}with positive $$c_1$$c1 and $$\epsilon $$∈ and $$C_0 \in U(2)$$C0?U(2). We show that the Heisenberg operator $$\hat{x}(t)$$x^(t) of the position operator converges to the asymptotic velocity operator $$\hat{v}_+$$v^+ so that $$\begin{aligned} \text{ s- }\lim _{t \rightarrow \infty } \mathrm{exp}\left( i \xi \frac{\hat{x}(t)}{t} \right) = \Pi _\mathrm{p}(U) + \mathrm{exp}(i \xi \hat{v}_+) \Pi _\mathrm{ac}(U) \end{aligned}$$s-limt??expi?x^(t)t=?p(U)+exp(i?v^+)?ac(U)provided that U has no singular continuous spectrum. Here $$\Pi _\mathrm{p}(U)$$?p(U) (resp., $$\Pi _\mathrm{ac}(U)$$?ac(U)) is the orthogonal projection onto the direct sum of all eigenspaces (resp., the subspace of absolute continuity) of U. We also prove that for the random variable $$X_t$$Xt denoting the position of a quantum walker at time $$t \in \mathbb {N}$$t?N, $$X_t/t$$Xt/t converges in law to a random variable V with the probability distribution $$\begin{aligned} \mu _V = \Vert \Pi _\mathrm{p}(U)\Psi _0\Vert ^2\delta _0 + \Vert E_{\hat{v}_+}(\cdot ) \Pi _\mathrm{ac}(U)\Psi _0\Vert ^2, \end{aligned}$$μV=??p(U)?0?2?0+?Ev^+(·)?ac(U)?0?2,where $$\Psi _0$$?0 is the initial state, $$\delta _0$$?0 the Dirac measure at zero, and $$E_{\hat{v}_+}$$Ev^+ the spectral measure of $$\hat{v}_+$$v^+.
['Akito Suzuki']
Asymptotic velocity of a position-dependent quantum walk