The quantum scattering limit for a regularized Wigner equation

We consider a regularized Wigner equation with an oscillatory kernel, the regularization acts in the space variable to damp high frequencies. The oscillatory kernel is directly derived from the Schrödinger equation with an oscillatory potential. The problem therefore contains three scales, ε the oscillation length, θ the regularization parameter, δ the potential lattice. We prove that the homogenized limit (as ε vanishes) of this equation is a scattering equation with discrete jumps. As δ vanishes, the discrete scattering kernel boils down to a standard regular scattering kernel. As θ vanishes we recover the quantum scattering operator with collisions preserving energy sphere.