Solution of the wave equation for the logarithmic potential with application to particle spectroscopy

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Abstract
We present an almost complete solution of the Schrödinger equation for a logarithmic potential. In particular we obtain two pairs of high‐energy asymptotic expansions of the boundstate eigenfunctions together with a corresponding expansion of the eigenvalue determined by the secular equation. We also obtain a pair of uniformly convergent solutions and a pair of uniform asymptotic expansions. Various properties of the solutions and eigenvalues are examined, including the scattering problem of the cut‐off 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 point out that the relevance of the logarithmic potential can be tested only if more than two consecutive energy levels are known. In a separate paper the methods outlined here are applied to quark‐confining potentials of the generalized power type.