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- W2238068711 abstract "We consider a position-dependent coined quantum walk on $$mathbb {Z}$$ and assume that the coin operator C(x) satisfies $$begin{aligned} Vert C(x) - C_0 Vert le c_1|x|^{-1-epsilon }, quad x in mathbb {Z}setminus {0} end{aligned}$$ with positive $$c_1$$ and $$epsilon $$ and $$C_0 in U(2)$$ . We show that the Heisenberg operator $$hat{x}(t)$$ of the position operator converges to the asymptotic velocity operator $$hat{v}_+$$ so that $$begin{aligned} text{ s- }lim _{t rightarrow infty } mathrm{exp}left( i xi frac{hat{x}(t)}{t} right) = Pi _mathrm{p}(U) + mathrm{exp}(i xi hat{v}_+) Pi _mathrm{ac}(U) end{aligned}$$ provided that U has no singular continuous spectrum. Here $$Pi _mathrm{p}(U)$$ (resp., $$Pi _mathrm{ac}(U)$$ ) 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 $$X_t$$ denoting the position of a quantum walker at time $$t in mathbb {N}$$ , $$X_t/t$$ converges in law to a random variable V with the probability distribution $$begin{aligned} mu _V = Vert Pi _mathrm{p}(U)Psi _0Vert ^2delta _0 + Vert E_{hat{v}_+}(cdot ) Pi _mathrm{ac}(U)Psi _0Vert ^2, end{aligned}$$ where $$Psi _0$$ is the initial state, $$delta _0$$ the Dirac measure at zero, and $$E_{hat{v}_+}$$ the spectral measure of $$hat{v}_+$$ ." @default.
- W2238068711 created "2016-06-24" @default.
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- W2238068711 date "2015-11-13" @default.
- W2238068711 modified "2023-10-14" @default.
- W2238068711 title "Asymptotic velocity of a position-dependent quantum walk" @default.
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- W2238068711 doi "https://doi.org/10.1007/s11128-015-1183-x" @default.
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