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. (Entropy 22(1):127, 2020).

我们研究整数晶格上的空间非均质量子游动（QW），我们将三个不同的硬币矩阵分别分配给正部分，负部分和原点。我们称它们为具有一个缺陷的两阶段QW。它们涵盖了已被深入研究的一缺陷和两相量子阱。定位是QW的最特征之一，具有一种缺陷的各种类型的两相QW表现出定位。此外，特征值的存在与本地化密切相关。在本文中，我们为特征值的存在获得了充要条件。我们的分析方法主要基于传递矩阵，传递矩阵是生成广义特征函数的有用工具。此外，我们明确推导了某些具有一个缺陷的两相QW的特征值，并用数字在单位圆上说明特征值的范围。我们的结果包括先前研究中的一些结果，例如Endo等。（熵22（1）：127，2020）。