Eigenvalues, absolute continuity and localizations for periodic unitary transition operators

The localization phenomenon for periodic unitary transition operators on a Hilbert space consisting of square summable functions on an integer lattice with values in a complex vector space, which is a generalization of the discrete-time quantum walks with constant coin matrices, are discussed. It is proved that a periodic unitary transition operator has an eigenvalue if and only if the corresponding unitary matrix-valued function on a torus has an eigenvalue which does not depend on the points on the torus. It is also proved that the continuous spectrum of the periodic unitary transition operators is absolutely continuous. As a result, it is shown that the localization happens if and only if there exists an eigenvalue, and the long time average of the transition probabilities coincides with the point-wise norm of the projection of the initial state to the direct sum of eigenspaces.