Spectral and scattering properties of quantum walks on homogenous trees of odd degree

TL;DR

This paper analyzes the spectral and scattering properties of quantum walks on homogenous trees of odd degree, establishing Mourre estimates and proving the spectrum is purely absolutely continuous except for finitely many eigenvalues.

Contribution

It introduces new Mourre estimates for quantum walks on trees and demonstrates the spectrum is fully absolutely continuous outside finitely many eigenvalues.

Findings

Spectrum covers the entire unit circle

Spectrum is purely absolutely continuous outside finite eigenvalues

Multiple free evolution operators can be used for wave operator proofs

Abstract

For unitary operators $U_0,U$ in Hilbert spaces ${\mathcal H}_0,{\mathcal H}$ and identification operator $J:{\mathcal H}_0\to{\mathcal H}$, we present results on the derivation of a Mourre estimate for $U$ starting from a Mourre estimate for $U_0$ and on the existence and completeness of the wave operators for the triple $(U,U_0,J)$. As an application, we determine spectral and scattering properties of a class of anisotropic quantum walks on homogenous trees of odd degree with evolution operator $U$. In particular, we establish a Mourre estimate for $U$, obtain a class of locally $U$-smooth operators, and prove that the spectrum of $U$ covers the whole unit circle and is purely absolutely continuous, outside possibly a finite set where $U$ may have eigenvalues of finite multiplicity. We also show that (at least) three different choices of free evolution operators $U_0$ are possible for the proof of the existence and completeness of the wave operators.