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

TL;DR

This paper analyzes the spectrum of a one-dimensional split-step quantum walk, establishing conditions for eigenvalues and demonstrating that certain bound states are exponentially localized and stable against perturbations.

Contribution

It applies a spectral mapping theorem to characterize eigenvalues and proves the robustness and exponential decay of bound states in the quantum walk.

Findings

Eigenvectors from birth eigenspaces decay exponentially at infinity.

Conditions for the absence or presence of eigenvalues near b1 1.

Birth eigenspaces are stable under 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.