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Pointwise Convergence of Solutions to the Schrödinger Equation on Manifolds

  • Xing Wang (a1) and Chunjie Zhang (a1)

Let $(M^{n},g)$ be a Riemannian manifold without boundary. We study the amount of initial regularity required so that the solution to a free Schrödinger equation converges pointwise to its initial data. Assume the initial data is in $H^{\unicode[STIX]{x1D6FC}}(M)$ . For hyperbolic space, the standard sphere, and the two-dimensional torus, we prove that $\unicode[STIX]{x1D6FC}>\frac{1}{2}$ is enough. For general compact manifolds, due to the lack of a local smoothing effect, it is hard to improve on the bound $\unicode[STIX]{x1D6FC}>1$ from interpolation. We managed to go below 1 for dimension ${\leqslant}$ 3. The more interesting thing is that, for a one-dimensional compact manifold, $\unicode[STIX]{x1D6FC}>\frac{1}{3}$ is sufficient.

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*Chunjie Zhang is the corresponding author.
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This work was supported by Zhejiang Provincial Natural Science Foundation (No. LY16A010013) and National Natural Science Foundation of China (Grant No. 11471288). The work was done when the two authors were working together at Johns Hopkins University.

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