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Eigenvalues of two-phase quantum walks with one defect in one dimension

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Abstract

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).

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Authors and Affiliations

  1. Mathematical Science Unit, Graduate School of Engineering Science, Yokohama National University, Hodogaya, Yokohama, 240-8501, Japan

    Chusei Kiumi

  2. Department of Information Systems Creation, Faculty of Engineering, Kanagawa University, Kanagawa, Yokohama, 221-8686, Japan

    Kei Saito

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  1. Chusei Kiumi
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  2. Kei Saito
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Correspondence to Chusei Kiumi.

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Kiumi, C., Saito, K. Eigenvalues of two-phase quantum walks with one defect in one dimension. Quantum Inf Process 20, 171 (2021). https://doi.org/10.1007/s11128-021-03108-x

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