Monotonic properties of the shift and penetration factors

TL;DR

This paper analyzes the derivatives of the shift and penetration factors in collision theory, establishing the sign of these derivatives for the repulsive Coulomb case and connecting them to Coulomb function properties.

Contribution

It provides definitive proofs for the signs of derivatives of shift and penetration factors with respect to energy, angular momentum, and charge in the Coulomb case, a previously unproven result.

Findings

Derivative of the shift factor with respect to energy is positive for repulsive Coulomb interactions.

Connected properties of Coulomb functions are investigated and characterized.

Provides theoretical insights into the behavior of collision parameters in Coulomb systems.

Abstract

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.