Classical Limit for the Varying-Mass Schrödinger Equation with Random Inhomogeneities

The varying-mass Schrödinger equation (VMSE) has been successfully applied to model electronic properties of semiconductor hetero-structures, for example, quantum dots and quantum wells. In this paper, we consider VMSE with small random heterogeneities, and derive a radiative transfer equation as its asymptotic limit. The main tool is to systematically apply the Wigner transform in the classical regime when the rescaled Planck constant , and expand the Wigner equation to proper orders of $\varepsilon\ll 1$. As a proof of concept, we numerically compute both VMSE and its limiting radiative transfer equation, and show that their solutions agree well in the classical regime.