Perturbation solutions of the semiclassical Wigner equation

TL;DR

This paper develops a perturbation method for solving the semiclassical Wigner equation using harmonic approximation and phase space analysis, providing efficient solutions and comparisons with classical approximations for anharmonic oscillators.

Contribution

It introduces a novel perturbation scheme based on a harmonic approximation and an innovative ansatz for the Wigner equation, improving solution efficiency in phase space.

Findings

The approximation is valid for specific initial data classes.

The method accurately computes energy density on caustics.

Results align well with classical Liouville solutions under certain conditions.

Abstract

We present a perturbation analysis of the semiclassical Wigner equation which is based on the interplay between configuration and phase spaces via Wigner transform. We employ the so-called harmonic approximation of the Schrodinger eigenfunctions for single-well potentials in configuration space, to construct an asymptotic expansion of the solution of the Wigner equation. This expansion is a perturbation of the Wigner function of a harmonic oscillator but it is not a genuine semiclassical expansion because the correctors depend on the semiclassical parameter. However, it suggests the selection of a novel ansatz for the solution of the Wigner equation, which leads to an efficient regular perturbation scheme in phase space. The validity of the approximation is proved for particular classes of initial data. The proposed ansatz is applied for computing the energy density of a a quartic oscillator on caustics (focal points). The results are compared with those derived from the so-called classical approximation whose principal term is the solution of Liouville equation with the same initial data. It turns out that the results are in good approximation when the coupling constant of the anharmonic potential has certain dependence on the semiclassical parameter.