Once you admit that the mass of a nonrelativistic quantum particle
may vary with its position (which may correspond, say, to its
motion in an external field in a crystal, etc), you may
immediately generalize the standard methods of solving
Schroedinger equation. The authors picked up the Lie-algebraic
approach (with Hamiltonians interpreted as Casimir operators) and
showed how one may construct exact solutions in single dimension
using representation theory and algebra so(2,1). Their conclusions
parallel the well known constant-mass classification scheme.
Slightly beyond this framework, a supersymmetry-related remark is
added on the existence of the so called intertwining relationship
between the two different ``partner" Hamiltonians.


MR2109635 Bagchi, B. ; Gorain, P. ; Quesne, C. ; Roychoudhury, R.
Effective-mass Schrödinger equation and generation of solvable
potentials. Czechoslovak J. Phys. 54 (2004), no. 10, 1019--1025.