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

Abstract

This paper begins by deriving the uniform bounds for the regular Coulomb wave function $F_{\ell ,\hat{I}\cdot }$ and its derivative $F_{\ell ,\hat{I}\cdot }^{\prime }$. We then examine detailed zero configurations of $F_{\ell ,\hat{I}\cdot }$ and $F_{\ell +1,\hat{I}\cdot }$, 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 $F_{\ell ,\hat{I}\cdot }$ with respect to parameters $\ell$ and $\hat{I}\cdot$, respectively. Furthermore, we expand our exploration to associated orthogonal polynomials, as well as the functions involving both $F_{\ell ,\hat{I}\cdot }$ and $F_{\ell ,\hat{I}\cdot }^{\prime }$. Finally, we explore the breakdown of the Sturm separation theorem by means of the zeros of associated orthogonal polynomials.

29 pages, 6 figures