Pointwise Convergence of Schrödinger Operators in Bessel Potential Spaces

Abstract

We study the pointwise convergence of solutions to the free Schrödinger equation with initial data in the Bessel potential spaces $L_s^p({\mathbb{R}}^n)$. We establish new sufficient regularity indices for pointwise convergence across the full range $1\le p<\infty$, and demonstrate via counterexamples that these indices are sharp for all $1\le p\le 2$ in one dimension, as well as for $p=1$ or $p$ large enough in higher dimensions. The proofs rely on the high-dimensional stationary phase method.

28 pages