Numerov and phase-integral methods for charmonium

This paper applies the Numerov and phase-integral methods to the stationary Schrödinger equation that studies bound states of charm anti-charm quarks. The former is a numerical method well suited for a matrix form of the second-order ordinary differential 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 Schrödinger 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.