Numerov and phase-integral methods for charmonium

TL;DR

This paper explores numerical and analytical methods, Numerov and phase-integral, to solve the stationary Schrödinger equation for charmonium, providing insights into bound states of charm quark pairs and clarifying conditions for higher-order quantization results.

Contribution

It applies and compares Numerov and phase-integral methods to charmonium, offering new evaluation of phase-integral quantization and clarifying previous higher-order results.

Findings

Numerov method aligns with previous results by Eichten et al.

Phase-integral method clarifies conditions for higher-order quantization.

The study enhances understanding of bound states in charm-anticharm systems.

Abstract

This paper applies the Numerov and phase-integral methods to the stationary Schrodinger equation that studies bound states of charm anti-charm quarks. The former is a numerical method well suited for a matrix form of second-order ordinary di erential equations, and can be applied whenever the stationary states admit a Taylor-series expansion. The latter is an analytic method that provides, in principle, even exact solutions of the stationary Schrodinger equation, and well suited for applying matched asymptotic expansions and higher order quantization conditions. The Numerov method is found to be always in agreement with the early results of Eichten et al., whereas an original evaluation of the phase-integral quantization condition clarifies under which conditions the previous results in the literature on higher-order terms can be obtained.