Sur les mesures de Wigner

Abstract

We study the properties of the Wigner transform for arbitrary functions in $L^2$ for hermitian kernels like the so-called density matrices. And we introduce some limits of these transforms for sequences of functions in $L^2$, limits that correspond to the semi-classical limit in Quantum Mechanics. The measures we obtain in this way, that we call Wigner measures, have various mathematical properties that we establish. In particular, we prove they satisfy, in linear situations (Schrödinger equations) or nonlinear ones (time-dependent Hartree equations), transport equations of Liouville or Vlasov type.