The stationary measure of a space-inhomogeneous three-state quantum walk on the line

Three-state quantum walks are quite different from two-state quantum walks. Following Endo and Konno (2013), one natural question is: What is the stationary measure of a space-inhomogeneous three-state quantum walk? In this paper, we consider a special space-inhomogeneous three-state quantum walk on the line, which we call the three-state Wojcik walk. We calculate its eigenvalues. And by using the SGF method introduced by Konno et al. (Quantum Inf Pocess 12:35–53, 2013), we obtain its stationary measure. We find that the measure decays exponentially with respect to position under some mild conditions; however, if the walk takes -1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-1$$\end{document} as its eigenvalue, the asymptotic behavior of the measure is independent of position, which contrasts sharply with that of two-state quantum walks.