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

摘要

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.