MCS-MP seminar: Laurent Laffleche

Speaker: Laurent Laffleche (ENS Lyon)

Title: Commutator Estimates and Quantitative Local Weyl's Law for
Schrödinger Operators with Non-Smooth Potentials

Abstract: In this talk, I will consider spectral projections on the
negative eigenvalues of Schrödinger operators with non-smooth
potentials, and present commutator estimates in Schatten norms uniform
in the Planck constant. The singularity of the potential allows to treat
in particular the case of the Hartree minimizers with Coulomb
interaction.

These estimates are taken as an hypothesis for the initial data in
several works about the derivation of the Vlasov equation from quantum
mechanics in the semiclassical limit. They can be seen as the quantum
analog of a regularity estimate for a characteristic function of the
phase space.

These estimates also allows to get quantitative versions of the local
and phase space Weyl laws in strong topologies.