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