Monotonic properties of the shift and penetration factors

We study derivatives of the shift and penetration factors of collision theory with respect to energy, angular momentum, and charge. Definitive results for the signs of these derivatives are found for the repulsive Coulomb case. In particular, we find that the derivative of the shift factor with respect to energy is positive for the repulsive Coulomb case, a long anticipated but heretofore unproven result. These results are closely connected to the properties of the sum of squares of the regular and irregular Coulomb functions; we also present investigations of this quantity.

Figure 1

The $\ell =0$ shift factor versus energy for the repulsive (blue dashed curve), uncharged (black solid curve), and attractive (red dot-dashed curve) cases. Additional details are provided in the text.