Eigenvalues of two-phase quantum walks with one defect in one dimension
arXiv:2010.08324 · doi:10.1007/s11128-021-03108-x
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).
arXiv comment11 pages 5 figures
References in corpus (12)
- A Quantum Random Walk Search Algorithm
- Spatial search by quantum walk
- Exploring Topological Phases With Quantum Walks
- Trapping a particle of a quantum walk on the line
- Quantum walks with an anisotropic coin I: spectral theory
- Quantum walks with an anisotropic coin II: scattering theory
- The stationary measure of a space-inhomogeneous quantum walk on the line
- Relation between two-phase quantum walks and the topological invariant
- A one-dimensional Hadamard walk with one defect
- Stationary measure for two-state space-inhomogeneous quantum walk in one dimension
- Stationary measures of three-state quantum walks on the one-dimensional lattice
- Eigenvalues of two-state quantum walks induced by the Hadamard walk
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- Strongly trapped space-inhomogeneous quantum walks in one dimension
- Localization in quantum walks with periodically arranged coin matrices
- An eigenfunction expansion formula for one-dimensional two-state quantum walks
- Spectral analysis of three-state quantum walks with general coin matrices
- Enhanced spreading in continuous-time quantum walks using aperiodic temporal modulation of defects