Generalized eigenfunctions for quantum walks via path counting approach
Mathematical Physics
2021-03-23 v2 math.MP
Abstract
We consider the time-independent scattering theory for time evolution operators of one-dimensional two-state quantum walks. The scattering matrix associated with the position-dependent quantum walk naturally appears in the asymptotic behavior at spatial infinity of generalized eigenfunctions. The asymptotic behavior of generalized eigenfunctions is a consequence of an explicit expression of the Green function associated with the free quantum walk. When the position-dependent quantum walk is a finite rank perturbation of the free quantum walk, we derive a kind of combinatorial constructions of the scattering matrix by counting paths of quantum walkers. We also mention some remarks on the tunneling effect.
Cite
@article{arxiv.2009.03498,
title = {Generalized eigenfunctions for quantum walks via path counting approach},
author = {Takashi Komatsu and Norio Konno and Hisashi Morioka and Etsuo Segawa},
journal= {arXiv preprint arXiv:2009.03498},
year = {2021}
}
Comments
21 pages, 1 figure