Near Sharp Strichartz estimates with loss in the presence of degenerate hyperbolic trapping

TL;DR

This paper establishes near-sharp Strichartz estimates with a loss for solutions to the Schrödinger equation on certain asymptotically Euclidean manifolds with degenerate hyperbolic trapping, extending previous results to broader settings.

Contribution

It constructs a semiclassical parametrix for long time scales and proves Strichartz estimates with a loss depending only on the dimension, not the degeneracy.

Findings

Strichartz estimates hold with loss depending only on dimension

Estimates are sharp up to an arbitrary small loss

Results contrast with local smoothing estimates depending on degeneracy

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 Schr\"odiner 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 $\beta>0$ loss. This is in contrast to \cite{ChWu-lsm}, 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.