Uniform bounds, zero separation and monotonicity for the regular Coulomb wave functions

TL;DR

This paper establishes uniform bounds, analyzes zero configurations, and studies monotonicity properties of the regular Coulomb wave functions and related orthogonal polynomials, extending previous results and exploring the limits of classical theorems.

Contribution

It introduces new uniform bounds for Coulomb wave functions, extends zero configuration analysis beyond previous parameter restrictions, and investigates the breakdown of Sturm separation theorem.

Findings

Derived uniform bounds for $F_{\, ext{ell},\, exteta}$ and its derivative.

Extended zero configuration analysis to broader parameter ranges.

Explored monotonicity of zeros and the failure of Sturm separation theorem.

Abstract

This paper begins by deriving the uniform bounds for the regular Coulomb wave function $F_{\ell,\eta}$ and its derivative $F_{\ell,\eta}'$. We then examine detailed zero configurations of $F_{\ell,\eta}$ and $F_{\ell+1,\eta}$, extending insights into the earlier work that was restricted to $\ell>-3/2$. Our investigation also includes an analysis of the monotonicity of the zeros of $F_{\ell,\eta}$ with respect to parameters $\ell$ and $\eta$, respectively. Furthermore, we expand our exploration to associated orthogonal polynomials, as well as the functions involving both $F_{\ell,\eta}$ and $F_{\ell,\eta}'$. Finally, we explore the breakdown of the Sturm separation theorem by means of the zeros of associated orthogonal polynomials.