The variable phase method is applied to the one-dimensional Schrödinger equation with position-dependent (effective) mass, to derive first-order differential equations for the transmission and reflection amplitudes, and bound-state energies, which are particularly convenient for numerical computations. When the mass and potential have the same asymptotics at both ends of the real line, the method also allows to prove a factorization property of the scattering matrix.

Sassoli de Bianchi, M. and Di Ventra, M. (1995). Differential equations and factorization property for the one-dimensional Schrödinger equation with position-dependent mass. Eur. J. Phys. 16, pp. 260-265. Doi: 10.1088/0143-0807/16/6/003.