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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
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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.
Keywords: pointwise convergence, Schrödinger operator, manifold, Strichartz estimate pointwise convergence, Schrödinger operator, manifold, Strichartz estimate
MSC Classifications: 35L05, 46E35, 42B37 show english descriptions Wave equation
Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems
Harmonic analysis and PDE [See also 35-XX]
35L05 - Wave equation
46E35 - Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems
42B37 - Harmonic analysis and PDE [See also 35-XX]
 

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