# Dynamical system induced by quantum walk

## TL;DR

This paper studies quantum walks on graphs with tails, showing the existence of stationary states, perfect transmission, and how adding tails affects scattering behavior, revealing fundamental properties of quantum transport on graphs.

## Contribution

It demonstrates the existence of stationary states and perfect transmission in quantum walks on graphs with tails, and characterizes scattering behavior with multiple tails, advancing understanding of quantum transport.

## Key findings

- Stationary states exist for any connected internal graph.
- Perfect transmission occurs in the long time limit.
- Adding more tails changes the scattering behavior to a local one-step scattering manner.

## Abstract

We consider the Grover walk model on a connected finite graph with two infinite length tails and we set an $\ell^\infty$-infinite external source from one of the tails as the initial state. We show that for any connected internal graph, a stationary state exists, moreover a perfect transmission to the opposite tail always occurs in the long time limit. We also show that the lower bound of the norm of the stationary measure restricted to the internal graph is proportion to the number of edges of this graph. Furthermore when we add more tails (e.g., $r$-tails) to the internal graph, then we find that from the temporal and spatial global view point, the scattering to each tail in the long time limit coincides with the local one-step scattering manner of the Grover walk at a vertex whose degree is $(r+1)$.