Multiple ordinary differential Schr\'{o}dinger equations are called
``solvable'' whenever they can be made equivalent, via a closed-form
change of variables, to the Gauss' (or confluent) hypergeometric
equation. A review provided in the first half of the paper is
complemented by the study of analogous mappings between
one-dimensional quantum models and the Heun ordinary differential
equation including also its confluent and generalized cases. These
mappings are of a generalized Natanzon type since in a preparatory
step an auxiliary nonlinear differential equation defines an
auxiliary function $y(x)$ of the coordinate $x$. The potential
itself is ultimately determined as a specific rational function of
$y$ while the coordinate itself is only available in an implicit
form of inverse function $x=x(y)$.



MR3066631 (Sent 2014-06-04) Batic, D.; Williams, R.; Nowakowski, M.
Potentials of the Heun class. J. Phys. A 46 (2013), no. 24, 245204,
21 pp. 81Q05 (34A34)