Near Sharp Strichartz Estimates with Loss in the Presence of Degenerate Hyperbolic Trapping

Abstract

We consider an n-dimensional spherically symmetric, asymptotically Euclidean manifold with two ends and a codimension 1 trapped set which is degenerately hyperbolic. By separating variables and constructing a semiclassical parametrix for a time scale polynomially beyond Ehrenfest time, we show that solutions to the linear Schrödinger equation with initial conditions localized on a spherical harmonic satisfy Strichartz estimates with a loss depending only on the dimension n and independent of the degeneracy. The Strichartz estimates are sharp up to an arbitrary β > 0 loss. This is in contrast to Christianson and Wunsch (Amer J Math, 2013), where it is shown that solutions satisfy a sharp local smoothing estimate with loss depending only on the degeneracy of the trapped set, independent of the dimension.