Limit distribution of a continuous-time quantum walk with a spatially 2-periodic Hamiltonian

Focusing on a continuous-time quantum walk on $\mathbb{Z}=\{0,\pm 1,\pm 2,\dots \}$, we analyze a probability distribution with which the quantum walker is observed at a position. The walker launches off at a localized state and its system is operated by a spatially periodic Hamiltonian. As a result, we see an asymmetric probability distribution. To catch a long-time behavior, we also try to find a long-time limit theorem and realize that the limit distribution holds a symmetric density function.