Limit theorem for continuous-time quantum walk on the line

Concerning a discrete-time quantum walk Xₜ⁽ᵈ⁾ with a symmetric distribution on the line, whose evolution is described by the Hadamard transformation, it was proved by the author that the following weak limit theorem holds: Xₜ⁽ᵈ⁾∕t→dx∕π(1-x²)√1-2x² as t→∞. The present paper shows that a similar type of weak limit theorem is satisfied for a continuous-time quantum walk Xₜ⁽ᶜ⁾ on the line as follows: Xₜ⁽ᶜ⁾∕t→dx∕π√1-x² as t→∞. These results for quantum walks form a striking contrast to the central limit theorem for symmetric discrete- and continuous-time classical random walks: Yₜ∕√t→e^-x²∕2dx∕√2π as t→∞. The work deals also with the issue of the relationship between discrete and continuous-time quantum walks. This topic, subject of a long debate in the previous literature, is treated within the formalism of matrix representation and the limit distributions are exhaustively compared in the two cases.