论文标题
两阶段量子步行的特征值，一个缺陷在一个维度上
Eigenvalues of two-phase quantum walks with one defect in one dimension
论文作者
论文摘要
我们在整数晶格上研究了空间量子量子步行（QW），我们分别为正部分，负部分和来源分别分配了三个不同的硬币矩阵。我们称它们为有一个缺陷的两阶段QW。它们涵盖了一台缺陷和两相QW，这些QW经过深入研究。本地化是QW的最有特征性的特性之一，并且具有一个缺陷的各种类型的两相QW。此外，特征值的存在与本地化密切相关。在本文中，我们获得了特征值的必要条件。我们的分析方法主要基于转移矩阵，这是一种有用的工具，用于生成广义特征功能。此外，我们明确地为具有一个缺陷的某些两相QW的类别得出了特征值，并说明了带有数字的单位圆圈上特征值的范围。我们的结果包括先前研究的一些结果，例如Endo等。 （2020）。
We study space-inhomogeneous quantum walks (QWs) on the integer lattice which we assign three different coin matrices to the positive part, the negative part, and the origin, respectively. We call them two-phase QWs with one defect. They cover one-defect and two-phase QWs, which have been intensively researched. Localization is one of the most characteristic properties of QWs, and various types of two-phase QWs with one defect exhibit localization. Moreover, the existence of eigenvalues is deeply related to localization. In this paper, we obtain a necessary and sufficient condition for the existence of eigenvalues. Our analytical methods are mainly based on the transfer matrix, a useful tool to generate the generalized eigenfunctions. Furthermore, we explicitly derive eigenvalues for some classes of two-phase QWs with one defect, and illustrate the range of eigenvalues on unit circles with figures. Our results include some results in previous studies, e.g. Endo et al. (2020).
