Complex Classical Motion in Potentials with Poles and Turning Points

Complex trajectories for Hamiltonians of the form H=pn+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.