Time operators for quantum walks

We study time operators for discrete-time quantum systems. Quantum walks are typical examples. We construct time operators for one-dimensional homogeneous quantum walks and determine their deficiency indices and spectra. Our time operators always have self-adjoint extensions. This is in contrast to the fact that time operators for continuous-time quantum systems generally have no self-adjoint extensions. The uniqueness of the extensions relates to the winding numbers corresponding to the system. If it is unique, its spectrum becomes a discrete set of real numbers, i.e., the time operator is quantized.