Eigenvalues of two-phase quantum walks with one defect in one dimension
- URL: http://arxiv.org/abs/2010.08324v2
- Date: Sat, 24 Oct 2020 10:44:37 GMT
- Title: Eigenvalues of two-phase quantum walks with one defect in one dimension
- Authors: Chusei Kiumi, Kei Saito
- Abstract summary: We study space-inhomogeneous quantum walks (QWs) on the integer lattice.
We call them two-phase QWs with one defect.
In this paper, we obtain a necessary and sufficient condition for the existence of eigenvalues.
- Score: 0.30458514384586394
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- 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. (2020).
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