  Schroedinger's equation with the attractive potential
  V(r) = -Z/(r^q+ b^q)^(1/q), Z > 0, b > 0, q >= 1, is shown, for general
values of the parameters Z and b, to be reducible to the confluent Heun
equation in the case q=1, and to the generalized Heun equation in 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. Detail solutions for both cases are provided.
