| 概要 | We will consider the Schr dinger equation,  , in  , with initial data  in potential spaces ) . Carleson proved that the solution converges, almost everywhere with respect to Lebesgue measure, to along the straight lines ) as  when . We improve this result in two ways. Firstly we show that the convergence holds everywhere apart from a set of Hausdorff dimension less than or equal to  when  , and that this is sharp.
Secondly we will prove that the convergence holds when the straight lines are replaced by continuously differentiable curves. This allows us to refine results of Sjgren--Torrea and Yajima for the quantum harmonic oscillator.
This is joint work with J.A. Barcel , J. Bennett, A. Carbery and S. Lee. |