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

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 $U_0$ existence completeness wave triple $(U,U_0,J)$. As an application, determine spectral scattering properties class anisotropic quantum walks homogenous trees odd degree with evolution $U$. In particular, establish $U$, obtain locally $U$-smooth operators, prove that spectrum covers whole unit circle is purely absolutely continuous, outside possibly finite set where may have eigenvalues multiplicity. We also show (at least) three different choices free are possible proof operators.