The localization phenomenon for periodic unitary transition operators on a Hilbert space consisting of square summable functions on an integer lattice with values in a finite-dimensional Hilbert space, which is a generalization of the discrete-time quantum walks with constant coin matrices, is discussed. It is proved that a periodic unitary transition operator has an eigenvalue if and only if the corresponding unitary matrix-valued function on a torus has an eigenvalue which does not depend on the points on the torus. It is also proved that the continuous spectrum of a periodic unitary transition operator is absolutely continuous. As a result, it is shown that the localization happens if and only if there exists an eigenvalue, and when there exists only one eigenvalue, the long-time limit of transition probabilities coincides with the point-wise norm of the projection of the initial state to the eigenspace. The results can be applied to certain unitary operators on a Hilbert space on a covering graph, called a topological crystal, over a finite graph. An analytic perturbation theory for matrices in several complex variables is employed to show the result about absolute continuity for periodic unitary transition operators.

中文翻译：

---

周期性酉转移算子的特征值、绝对连续性和局部化

讨论了希尔伯特空间上周期性酉跃迁算子的局部化现象，该空间由整数格上的平方可和函数组成，其值在有限维希尔伯特空间中，是离散时间量子游走与常数硬币矩阵的推广。证明了一个周期酉转移算子有一个特征值当且仅当一个环面上对应的酉矩阵值函数有一个不依赖于环面上点的特征值。还证明了周期酉跃迁算子的连续谱是绝对连续的。结果表明，当且仅当存在一个特征值时，本地化才会发生，并且当仅存在一个特征值时，转移概率的长期限制与初始状态到特征空间的投影的逐点范数一致。结果可以应用于有限图上覆盖图（称为拓扑晶体）上希尔伯特空间上的某些酉算子。采用多复变量矩阵的解析摄动理论，给出了周期酉跃迁算子绝对连续性的结果。