# Resolvent Methods for Quantum Walks with an Application to a Thue-Morse Quantum Walk

## TL;DR

This paper explores resolvent methods for analyzing one-dimensional quantum walks, demonstrating their application to a Thue-Morse quantum walk, and providing a framework to connect eigenvalue problems with dynamical behavior.

## Contribution

It introduces a mathematical approach using resolvent methods to study inhomogeneous quantum walks and applies it specifically to a Thue-Morse quantum walk example.

## Key findings

- Method translates eigenvalue equations into dynamical estimates
- Application to Thue-Morse quantum walk illustrates the approach
- Provides a framework for analyzing spatially inhomogeneous quantum walks

## Abstract

In this expository note, we discuss spatially inhomogeneous quantum walks in one dimension and describe a genre of mathematical methods that enables one to translate information about the time-independent eigenvalue equation for the unitary generator into dynamical estimates for the corresponding quantum walk. To illustrate the general methods, we show how to apply them to a 1D coined quantum walk whose coins are distributed according to an element of the Thue--Morse subshift.