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.

中文翻译：

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量子行走的时间算子

我们研究离散时间量子系统的时间算子。量子行走就是典型的例子。我们为一维齐次量子行走构造时间算子并确定它们的缺陷指数和光谱。我们的时间算子总是有自伴随扩展。这与连续时间量子系统的时间算符通常没有自伴随扩展的事实形成对比。扩展的唯一性与系统对应的绕组数有关。如果它是唯一的，它的频谱就变成了一组离散的实数，即时间算子被量化了。