Spectral and Scattering Properties of Quantum Walks on Homogenous Trees of Odd Degree

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.