HOMOGENIZATION LIMITS AND WIGNER TRANSFORMS

We present a theory for carrying out homogenization limits for quadrat ic functions (called ''energy densities'') of solutions of initial val ue problems (IVPs) with anti-self-adjoint (spatial) pseudo-differentia l operators (PDOs). The approach is based on the introduction of phase space Wigner (matrix) measures that are calculated by solving kinetic equations involving the spectral properties of the PDO. The weak limi ts of the energy densities are then obtained by taking moments of the Wigner measure. The very general theory is illustrated by typical exam ples like (semi)classical limits of Schrodinger equations (with or wit hout a periodic potential), the homogenization limit of the acoustic e quation in a periodic medium, and the classical limit of the Dirac equ ation. (C) 1997 John Wiley & Sons, Inc.