Energy Dependence

The scattering functions, $T_E(b)$ and $\theta_E(b)$, showed a fractal behavior when calculated at a given energy, $E = \frac{1}{2} m v_0^2$. Recall that the total energy, $E = K + V$, is a constant of motion whose value is specified by the initial conditions.

The scattering functions are expected to depend on the energy in a non trivial way. For instance, trajectories are excluded from some $x-y$ regions under the potential mounds for $E < \frac{1}{2} k b^2$ (maximum of the mounds). Another way to think about this energy dependence is to recall that the scattering trajectories are influenced by the proximity of unstable periodic orbits. But different such orbits exist at different energies as was seen previously, resulting in an energy dependence of the scattering functions.

This points to the necessity of calculating $T_E(b)$ and $\theta_E(b)$ as functions of both $E$ and $b$. It also requires to display the results in the form of 3-dimension surfaces or color images. Expect complex structures!