Semiclassical limit of orthonormal Strichartz estimates on scattering manifolds

Abstract

We study a quantum and classical correspondence related to the Strichartz estimates. First we consider the orthonormal Strichartz estimates on manifolds with ends. Under the nontrapping condition we prove the global-in-time estimates on manifolds with asymptotically conic ends or with asymptotically hyperbolic ends. Then we show that, for a class of pseudodifferential operators including the Laplace-Beltrami operator on the scattering manifolds, such estimates imply the global-in-time Strichartz estimates for the kinetic transport equations in the semiclassical limit. As a byproduct we prove that the existence of a periodic stable geodesic breaks the orthonormal Strichartz estimates. In the proof we do not need any quasimode. As an application we show the small data scattering for the cutoff Boltzmann equation on nontrapping scattering manifolds.