Abstract view
Pointwise convergence of solutions to the Schrödinger equation on manifolds


Xing Wang,
Department of Mathematics, Wayne State University, Detroit, Michigan 48202, USA
Chunjie Zhang,
Department of Mathematics, Hangzhou Dianzi University , Hangzhou, 310018, China
Abstract
Let $(M^n,g)$ be a Riemannian manifold without
boundary. We study the amount of initial regularity is required
so that the solution to free Schrödinger equation converges
pointwise to its initial data. Assume the initial data is in
$H^\alpha(M)$. For Hyperbolic Space, standard Sphere and the
2 dimensional Torus, we prove that $\alpha\gt \frac{1}{2}$ is enough.
For general compact manifolds, due to lacking of local smoothing
effect, it is hard to beat the bound $\alpha\gt 1$ from interpolation.
We managed to go below 1 for dimension $\leq 3$. The more interesting
thing is that, for 1 dimensional compact manifold, $\alpha\gt \frac{1}{3}$
is sufficient.