Friedel oscillations from the Wigner-Kirkwood distribution in half infinite matter

The Friedel oscillations are interference effects on the density of a half infinite system, close to the surface. They are also present in the Wigner transform of the density matrix and the components of the momentum are respectively parallel and perpendicular to the surface of the medium infinite in x and y directions. For the ramp potential and the harmonic oscillator, the Wigner distribution depends only on the classical hamiltonian; therefore, in the general case a convenient choice of variables are in some combinations. In one example the Wigner density depends only on the classical hamiltonian as the integral Airy function and is written as an inverse Laplace transform. This paper presents a distribution representation of the density matrix which must contain the effects of the Friedel oscillations. In order to gain more insight into the validity of this expansion, it is derived in the same way as the low temperature expansion of the Fermi function