Explicit solution of the wave equation for arbitrary power potentials with application to charmonium spectroscopy

Abstract

We present an explicit and almost complete series solution of the Schrödinger equation for an arbitrary quark‐confining power potential with or without a weak Coulomb component or other corrections. In particular, we derive two pairs of high‐energy asymptotic expansions of the bound‐state eigenfunctions together with a corresponding expansion of the eigenvalue determined by the secular equation. We also obtain a pair of uniformly convergent expansions and discuss other types of solutions. Various properties of the solutions and eigenvalues are examined including the scattering problem of the cutoff potential and the behavior of Regge trajectories. Finally, the relevance of these investigations to the spectroscopy of heavy quark composites is discussed. In particular, we derive approximate expressions for leptonic decay rates. Examples are given to demonstrate the usefulness of these results for theoretical discussion and as alternatives for numerical integration techniques. A subsequent paper will deal with the normalization of the bound‐state wavefunctions and the corresponding derivation of explicit series expressions for certain decay rates.