  Based on recent results on quasi-exactly solvable Schrodinger equations, we
review a new phenomenological potential class lately reported. In the present
paper we consider the quantum differential equations resulting from position
dependent mass (PDM) particles. We focus on the PDM version of the hyperbolic
potential $V(x) = {a}~\text{sech}^2x + {b}~\text{sech}^4x$, which we address
analytically with no restrictions on the parameters and the energies. This is
the celebrated Manning potential, a double-well widely known in molecular
physics, until now not investigated for PDM. We also evaluate the PDM version
of the sixth power hyperbolic potential $V(x) =
{a}~{\text{sech}^6x}+b~{\text{sech}^4x}$ for which we could find exact
expressions under some special settings. Finally, we address a triple-well case
$V(x) = {a}~{\text{sech}^6x}+b~{\text{sech}^4x}+c~\text{sech}^2x$ of particular
interest for its connection to the new trends in atomtronics. The PDM
Schrodinger equations studied in the present paper yield analytical
eigenfunctions in terms of local Heun functions in its confluents forms. In all
the cases PDM particles are more likely tunneling than ordinary ones. In
addition, a merging of eigenstates has been observed when the mass becomes
nonuniform.
