# Quantum walks with an anisotropic coin I: spectral theory

## TL;DR

This paper conducts a spectral analysis of quantum walks with anisotropic coins, revealing their spectral properties and introducing new commutator methods for unitary operators in a two-Hilbert space framework.

## Contribution

It provides a comprehensive spectral analysis of anisotropic quantum walks and introduces novel commutator techniques for unitary operators in a two-Hilbert space setting.

## Key findings

- Determined the essential spectrum of the evolution operator.
- Proved the discreteness of eigenvalues outside thresholds.
- Established the absence of singular continuous spectrum.

## Abstract

We perform the spectral analysis of the evolution operator U of quantum walks with an anisotropic coin, which include one-defect models, two-phase quantum walks, and topological phase quantum walks as special cases. In particular, we determine the essential spectrum of U, we show the existence of locally U-smooth operators, we prove the discreteness of the eigenvalues of U outside the thresholds, and we prove the absence of singular continuous spectrum for U. Our analysis is based on new commutator methods for unitary operators in a two-Hilbert spaces setting, which are of independent interest.