Complex Classical Motion in Potentials with Poles and Turning Points

Complex trajectories for Hamiltonians of the form H = p n + V ( x )are studied. For n = 2 , time‐reversal symmetry prevents trajectories from crossing. However, for n > 2 trajectories may indeed cross, and as a result, the complex trajectories for such Hamiltonians have a rich and elaborate structure. In past work on complex classical trajectories, it has been observed that turning points act as attractors; they pull on complex trajectories and make them veer toward the turning point. In this paper, it is shown that the poles of V ( x ) have the opposite effect—they deflect and repel trajectories. Moreover, poles shield and screen the effect of turning points.