Localization for a one-dimensional split-step quantum walk with bound states robust against perturbations

Abstract

For given two unitary and self-adjoint operators on a Hilbert space, a spectral mapping theorem was proved in \cite{HiSeSu}. In this paper, as an application of the spectral mapping theorem, we investigate the spectrum of a one-dimensional split-step quantum walk. We give a criterion for when there is no eigenvalues around $\pm 1$ in terms of a discriminant operator. We also provide a criterion for when eigenvalues $\pm 1$ exist in terms of birth eigenspaces. Moreover, we prove that eigenvectors from the birth eigenspaces decay exponentially at spatial infinity and that the birth eigenspaces are robust against perturbations.

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Cited by in corpus (13)

• Quantum walks with an anisotropic coin I: spectral theory
• Generalized eigenfunctions and scattering matrices for position-dependent quantum walks
• Weak limit theorem for a one-dimensional split-step quantum walk
• The Witten Index for 1D Supersymmetric Quantum Walks with Anisotropic Coins
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• Unitary equivalence classes of split-step quantum walks
• Supersymmetric quantum walks with chiral symmetry
• The bulk-edge correspondence for the split-step quantum walk on the one-dimensional integer lattice
• Spectral analysis for a multi-dimensional split-step quantum walk with a defect
• Index Theorems for One-dimensional Chirally Symmetric Quantum Walks with Asymptotically Periodic Parameters
• The Witten index for one-dimensional split-step quantum walks under the non-Fredholm condition
• The Witten Index for One-dimensional Non-unitary Quantum Walks with Gapless Time-evolution
• An index theorem for split-step quantum walks