Pointwise Convergence of Solutions to the Schrödinger Equation on Manifolds

Wang, Xing ; Zhang, Chunjie

Canadian journal of mathematics, Tome 71 (2019) no. 4, pp. 983-995

Résumé

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.

Détail

DOI : 10.4153/CJM-2018-001-4

Mots-clés : pointwise convergence, Schrödinger operator, manifold, Strichartz estimate

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