Soft-core Coulomb potentials and Heun’s differential equation

Schrödinger’s equation with the attractive potential V(r)=−Z/(rq+βq)1/q, Z>0, β>0, q≥1, is shown, for general values of the parameters Z and β, to be reducible to the confluent Heun equation in the case q=1 and to the generalized Heun equation in the case q=2. In a formulation with correct asymptotics, the eigenstates are specified a priori up to an unknown factor. In certain special cases, this factor becomes a polynomial. The asymptotic iteration method is used either to find the polynomial factor and the associated eigenvalue explicitly, or to construct accurate approximations for them. Detailed solutions for both cases are provided.