Spectral Properties of Quantum Walks on Rooted Binary Trees
arXiv:1307.2772 · doi:10.1007/s10955-014-0950-x
Abstract
We define coined Quantum Walks on the infinite rooted binary tree given by unitary operators on an associated infinite dimensional Hilbert space, depending on a unitary coin matrix , and study their spectral properties. For circulant unitary coin matrices , we derive an equation for the Carathéodory function associated to the spectral measure of a cyclic vector for . This allows us to show that for all circulant unitary coin matrices, the spectrum of the Quantum Walk has no singular continuous component. Furthermore, for coin matrices which are orthogonal circulant matrices, we show that the spectrum of the Quantum Walk is absolutely continuous, except for four coin matrices for which the spectrum of is pure point.
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