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The varying-mass Schr\"odinger equation (VMSE) has been successfully appliedto model electronic properties of semiconductor hetero-structures, for example,quantum dots and quantum wells. In this paper, we consider VMSE with smallrandom heterogeneities, and derive a radiative transfer equation as itsasymptotic limit. The main tool is to systematically apply the Wigner transformin the classical regime when the rescaled Planck constant $\epsilon\ll 1$, andexpand the Wigner equation to proper orders of $\epsilon$. As a proof ofconcept, we numerically compute both VMSE and its limiting radiative transferequation, and show that their solutions agree well in the classical regime.

Classical limit for the varying-mass Schr\"odinger equation with random inhomogeneities