Generalized eigenfunctions for quantum walks via path counting approach

TL;DR

This paper develops a path counting approach to analyze generalized eigenfunctions and scattering matrices in one-dimensional quantum walks, revealing their asymptotic behavior and effects like tunneling.

Contribution

It introduces a combinatorial method to construct the scattering matrix for finite rank perturbations of free quantum walks using path counting techniques.

Findings

Explicit expression for the Green function of free quantum walks

Path counting method for scattering matrix construction

Remarks on tunneling effects in quantum walks

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.