**Relation to an absorption problem**

Let the reflection amplitude at time *n* be *<sup>γ</sup>*˜*n*(*z*) := *L*|<sup>Φ</sup>*n*(−<sup>1</sup>) with *z* = *eiξ* . We can see that *<sup>γ</sup>*˜*n*(*z*) is rewritten by using *U* as follows:

$$\begin{split} z^{-1}\tilde{\gamma}\_{n+1}(z) &= \langle \delta\_{(-1;\mathbb{L})}, \mathcal{U}^{\ell}\delta\_{(0;\mathbb{R})}\rangle + \langle \delta\_{(-1;\mathbb{L})}, \mathcal{U}^{\ell^2}\delta\_{(0;\mathbb{R})}\rangle z \\ &+ \langle \delta\_{(-1;\mathbb{L})}, \mathcal{U}^{\ell^3}\delta\_{(0;\mathbb{R})}\rangle z^2 + \dots + \langle \delta\_{(-1;\mathbb{L})}, \mathcal{U}^{\prime n+1}\delta\_{(0;\mathbb{R})}\rangle z^n \end{split}$$

The first term is the amplitude that the inflow at time *n* cannot penetrate into Γ*M*; the *m*-th term is the amplitude that the inflow at time *n* − (*m* − 1) penetrates into Γ*M* and escapes Γ*M* from 0 side at time *n*. Therefore, each term corresponds to the "absorption" amplitude to −1 with the absorption walls −1 and *M* with the initial state *<sup>δ</sup>*(0;*<sup>R</sup>*). Then

**Remark 1.** *The reflection amplitude L*|Φ ∞(−<sup>1</sup>) = lim*n*<sup>→</sup> ∞ *<sup>γ</sup>*˜*n*(*z*) *coincides with the generating function of the absorption amplitude to* −1 *with respect to time n while the transmitting amplitude R*|Φ ∞(*M*) = lim*n*<sup>→</sup> ∞ *<sup>τ</sup>*˜*n*(*z*) *coincides with the generating function of the absorption amplitude to M with respect to time n.*

Put *γn* := |*<sup>δ</sup>*(−1;*L*), *U<sup>n</sup> <sup>δ</sup>*(0;*R*)|<sup>2</sup> and *τn* := |*<sup>δ</sup>*(*<sup>M</sup>*;*<sup>R</sup>*), *U<sup>n</sup> <sup>δ</sup>*(0;*R*)|<sup>2</sup> which are the absorption/ first hitting probabilities at positions −1 and *M*, respectively, starting from (0 : *<sup>R</sup>*). From the above observation, for example, we can express the *m*-th moments of the absorption/hitting times to −1 and *M* as follows:

$$\sum\_{n\geq 1} n^m \gamma\_n = \int\_0^{2\pi} \overline{\langle L|\Phi\_{\infty}(-1)\rangle} \left(-i\frac{\partial}{\partial \xi}\right)^m \langle L|\Phi\_{\infty}(-1)\rangle \frac{d\xi}{2\pi'}\tag{3}$$

$$\sum\_{n\geq 1} n^m \tau\_n = \int\_0^{2\pi} \overline{\langle R | \Phi\_\infty(M) \rangle} \left( -i \frac{\partial}{\partial \overline{\xi}} \right)^m \langle R | \Phi\_\infty(M) \rangle \frac{d\overline{\xi}}{2\pi}.\tag{4}$$

### **Relation to Scattering of quantum walk**

The stationary state Φ ∞ is a generalized eigenfunction of *U* in ∞(<sup>Z</sup>; <sup>C</sup><sup>2</sup>). The scattering matrix naturally appears in Φ ∞ (see [5]). In the time independent scattering theory, the inflow can be considered as the incident "plane wave", and the impurity causes the scattered wave by transmissions and reflections. Thus, we can see the transmission coefficient and the reflection coefficient in Φ ∞(*x*) for *x* ∈ Z \ Γ*M*. For studies of a general theory of scattering, we also mention the recent work by Tiedra de Aldecoa [12].
