Implication of Airy Function Zeros for the Heavy Meson Mass Spectra

As the Airy function is the exact solution of the non-relativistic quantum mechanical wave equation, we look into the correspondence of the Airy function zeros to the linearly confining heavy meson mass spectroscopy. In particular, we analyze the contribution of the linearly confining potential to the 1S and 2S mass gaps of heavy meson systems consisting of bottom and charm quarks. Our derivation of mass scaled Airy function zeros indicates the difference of the confining contribution for the bottomonium and charmonium spectroscopy. Considering the size of the heavy meson systems, namely that the bottomonium radius is smaller than the charmonium radius, we find that the confinement effect is more significant in the larger system than in the smaller system, while the Coulomb effect contributes more significantly to the smaller system than to the larger system.