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.

Revised version (with new title and more general results) to appear in Annales Henri Poincaré

References in corpus (2)

• Quantum walks with an anisotropic coin I: spectral theory
• Quantum walks with an anisotropic coin II: scattering theory