Abstract
We derive the high frequency limit of the Helmholtz equations in
terms of quadratic observables. We prove that it can be written as a
stationary Liouville equation with source terms. Our method is based
on the Wigner Transform, which is a classical tool for evolution
dispersive equations. We extend its use to the stationary case after
an appropriate scaling of the Helmholtz equation. Several specific
difficulties arise here; first, the identification of the source term
(which does not share the quadratic aspect) in the limit, then, the
lack of $L^{2}$ bounds which can be handled with homogeneous
Morrey-Campanato estimates, and finally the problem of uniqueness
which, at several stage of the proof, is related to outgoing
conditions at infinity.