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In this talk, we discuss the complex semiclassical method applied to dynamical tunneling in continuous time systems focusing on the case that the stable-unstable manifold guided tunneling(SUMGT) dominates the tunneling process. According to Miller's prescription, we first explain how to apply the complex semiclassical method to continuous time systems in the case of scattering. Next we give the overview of SUMGT clarifying the difference from the instanton type tunneling and discuss what is the major problem to handle SUMGT in continuous time systems. The complex stable and unstable manifolds are characteristic manifolds in the way that the singularities of a trajectory on them behave in a critical manner. Namely, a part of singularities are lost for the trajectories on the stable and unstable manifolds due to their divergence. Those characteristic manifolds guides the complex tunneling trajectories in the SUMGT scenario and govern their behavior thereby remarkably affecting the tunneling mechanism and the resultant tunneling probability. Therefore we must take special care in the calculation of the complex trajectories contributing SUMGT. We introduce the Melnikov method as a practical method to estimate the tunneling rate in the case that SUMGT works in the tunneling process. Taking a periodically perturbed burrer potential for example, we explain how the competition between instanton and SUMGT occurs and how to decide the winner of the competition which dominates the tunneling process. Actually the tunneling rate markedly changes with change of perturbation frequency reflecting the change of dominant mechanism between instanton and SUMGT. Finally taking a periodically perturbed rounded off step potential, for which the instanton type tunneling is substantially restricted, we discuss characteristics of non-instanton tunneling. Especially we focus on the following problems: 1)What is the remarkable feature of the non-instanton tunneling. 2) How the SUMGT scenario works in this case. 3) If it works, what plays the role of the stable and unstable manifolds in this case. 4) How the tunneling rate changes with change of the perturbation frequency. |
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