Localization for a one-dimensional split-step quantum walk with bound states robust against perturbations
Toru Fuda, Daiju Funakawa, Akito Suzuki · Journal of Mathematical Physics · 2018
For given two unitary and self-adjoint operators on a Hilbert space, a spectral mapping theorem was proved in the work of Higuchi et al. (e-print arXiv:1506.06457) [see also E. Segawa and A. Suzuki, Quantum Stud.: Math. Found. 3, 11 (2016)]. 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 are no eigenvalues around ±1 in terms of a discriminant operator. We also provide a criterion for when eigenvalues ±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.