NEAR SHARP STRICHARTZ ESTIMATES WITH LOSS IN THE PRESENCE OF DEGENERATE HYPERBOLIC TRAPPING

Abstract. We consider an n-dimensional spherically symmetric, asymptot-ically 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 con-ditions 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 [CW11], 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. 1