Smoothed Wigner transforms and homogenization of wave propagation

The Wigner transform (WT) is a well known quadratic transform, mapping a wavefunction to a position-wavenumber quasi-density. WTs are used in the asymptotic treatment of semiclassical and other wave problems. The smoothed WT (SWT) method is a regularization and extension of WT-based approaches for the homogenization of wave propagation, which decouples homogenization from asymptotics: exact equations for the evolution of a given problem at coarse scale can be obtained, instead of merely looking at small parameter limits. In this work we present the derivation of the SWT equations