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Focusing on a continuous-time quantum walk on $\mathbb{Z}=\left\{0,\pm 1,\pm2,\ldots\right\}$, we analyze a probability distribution with which the quantumwalker is observed at a position. The walker launches off at a localized stateand its system is operated by a spatially periodic Hamiltonian. As a result, wesee an asymmetric probability distribution. To catch a long-time behavior, wealso try to find a long-time limit theorem and realize that the limitdistribution holds a symmetric density function.

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