Asymptotic velocity of a position-dependent quantum walk

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We consider a position-dependent coined quantum walk on Z and assume that the coin operator C(x) satisfies C(x) - C₀ ≤ c₁|x|⁻¹⁻ε, x ∈ Z with positive c₁ and ε and C₀ ∈ U(2). We show that the Heisenberg operator x(t) of the position operator converges to the asymptotic velocity operator v+ so that s-ₜ → ∞ exp(i ξ x(t)/t) = Π p(U) + exp(i ξ v+) Π ac(U) provided that U has no singular continuous spectrum. Here Π p(U) (resp. Π 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 Xₜ denoting the position of a quantum walker at time t ∈ N, Xₜ/t converges in law to a random variable V with the probability distribution μᵥ = Π p(U)Ψ₀²δ₀ + E v+(·) Π ac(U)Ψ₀², where Ψ₀ is the initial state, δ₀ the Dirac measure at zero, and E v+ the spectral measure of v+.