Discrete-to-Continuum Approach for the Analytic Continuation of One-Particle Propagator on the Circle

Despite the simplicity of one-particle dynamics, explicit expressions for the one-dimensional propagator on a circle suitable to numerical evaluation are surprisingly lacking -- not only in the presence of potentials but even in the free case. Using a lattice regularization of the circle, we derive finite expressions for the free discrete propagator through an algebraic approach, aiming to provide physical insight into the readout of a digital quantum simulation. Moreover, these expressions allow for the reconstruction of the propagator in the continuous circle limit, which exhibits in the free case a peculiar non-analytic behavior in its transition between irrational and rational times. The latter propagator yields a finite analytic continuation of the corresponding elliptic theta function at the locus of essential singularities for real times, achieved through the introduction of a $\sigma$ distribution -- the ``square-root'' of the Dirac delta. We also show that the well-known infinite line limit is consistently recovered within this approach. In addition, we apply these results by studying numerically the dynamics of wave packets in cosine and random potentials. At early simulation times, we observe evidence of the semi-classical limit, where the probability density maximum follows the minimum of the propagator phase.