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

For unitary operators U 0 , U in Hilbert spaces H 0 , H and identification operator J : H 0 → 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.