High-frequency averaging in the semi-classical singular Hartree equation (Q2857023)

The authors investigate the asymptotic behavior of solutions to semi-classical Schrödinger equations with nonlinearities of Hartree type. For a weakly nonlinear scaling, they show the validity of the WKB-analysis when the potential in the nonlinearity is singular around the origin. No new resonant wave is created in their model, this phenomenon being inhibited by the nonlinearity. The nonlocal nature of this latter leads the author to build his result on a high-frequency averaging effects. In the proof, he makes use of the Wiener algebra and the space of square-integrable functions.