# Eigenvalues of quantum walk induced by recurrence properties of the underlying birth and death process: application to computation of an edge state

## TL;DR

This paper links the spectral properties of quantum walks to the recurrence behavior of underlying birth and death processes, providing a new method to compute edge states in topological phases.

## Contribution

It introduces a novel connection between recurrence properties of birth and death processes and the spectral analysis of quantum walks, enabling easier computation of edge states.

## Key findings

- Point spectrum exists if the underlying walk is not null recurrent.
- Recurrence properties determine the spectral features of quantum walks.
- A simple computational method for edge state dispersion relations is proposed.

## Abstract

In this paper, we consider an extended coined Szegedy model and discuss the existence of the point spectrum of induced quantum walks in terms of recurrence properties of the underlying birth and death process. We obtain that if the underlying random walk is not null recurrent, then the point spectrum exists in the induced quantum walks. As an application, we provide a simple computational way of the dispersion relation of the edge state part for the topological phase model driven by quantum walk using the recurrence properties of underlying birth and death process.