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arXiv · 2304.05247

Strichartz estimates for the Schr\"odinger equation on negatively curved compact manifolds

Matthew D. Blair·Xiaoqi Huang·Christopher D. Sogge
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Abstract

We obtain improved Strichartz estimates for solutions of the Schr\"odinger equation on negatively curved compact manifolds which improve the classical universal results results of Burq, G\'erard and Tzvetkov [11] in this geometry. In the case where the spatial manifold is a hyperbolic surface we are able to obtain no-loss $L^{q_c}_{t,x}$-estimates on intervals of length $\log \lambda\cdot \lambda^{-1} $ for initial data whose frequencies are comparable to $\lambda$, which, given the role of the Ehrenfest time, is the natural analog of the universal results in [11]. We are also obtain improved endpoint Strichartz estimates for manifolds of nonpositive curvature, which cannot hold for spheres.

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Matthew D. Blair, Xiaoqi Huang, Christopher D. Sogge. 2023-04-11. Strichartz estimates for the Schr\"odinger equation on negatively curved compact manifolds. https://arxiv.org/abs/2304.05247

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