**Proposition 1** ([14])**.**

*1. This quantum walk converges to a stationary state in the following meaning:*

$$\exists \lim\_{n \to \infty} e^{i(n+1)\xi} \Psi\_n(x) =: \Phi\_{\infty}(x).$$

*2. This stationary state is a generalized eigenfunction satisfying*

$$
\mathcal{U}\Phi\_{\otimes} = e^{-i\vec{\zeta}}\Phi\_{\otimes}.
$$
