Analytical approximations to the l-wave solutions of the Schrödinger equation with a hyperbolic potential

The bound-state solutions of the Schrödinger equation for a hyperbolic potential with the centrifugal term are presented approximately. It is shown that the solutions can be expressed by the hypergeometric function ₂F₁(a, b; c; z). To show the accuracy of our results, we calculate the energy levels numerically for arbitrary quantum numbers n and l. It is found that the results are in good agreement with those obtained by other methods for short-range potential. Two special cases for l = 0 and σ = 1 are also studied briefly.