Eigenvalues, absolute continuity and localizations for periodic unitary transition operators
Eigenvalues, absolute continuity and localizations for periodic unitary transition operators
复制标题
DOI：
10.1142/s0219025719500115
复制
发表时间：
2019-06-01
影响因子：
0.9
通讯作者：
Tate, Tatsuya
中科院分区：
文献类型：
--
作者：
Tate, Tatsuya
讨论了由整数格上的平方可和函数组成的Hilbert空间上的周期幺正转移算子的局部化现象，它是具有常数硬币矩阵的离散时间量子游动的推广.证明了周期酉转移算子有特征值当且仅当环面上相应的酉矩阵值函数有不依赖于环面上点的特征值.证明了周期酉转移算子的连续谱是绝对连续的。结果表明，局部化发生的充要条件是存在一个本征值，并且当只有一个本征值时，转移概率的长时间极限与初始状态在本征空间的投影的逐点范数一致.这些结果可以应用于有限图上覆盖图（称为拓扑晶体）上Hilbert空间上的某些酉算子。利用多复变矩阵的解析扰动理论，给出了周期酉转移算子的绝对连续性结果。
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.