On the size of divergence sets for the Schrödinger equation with radial data

We consider the Schrödinger equation iδ_tu + Δu = 0 with initial data in H^s(ℝⁿ). A classical problem is to identify the exponents s for which u(·,t) converges almost everywhere to the initial data as t tends to zero. In one spatial dimension, Carleson proved that the convergence is guaranteed when s = 1, 4 and Dahlberg and Kenig proved that divergence can occur on a set of nonzero Lebesgue measure when s < 14. In higher dimensions Prestini deduced the same conclusions when restricting attention to radial data. We refine this by proving that the Hausdorff dimension of the divergence sets can be at most n - 1/2 for radial data in H^1/4(ℝⁿ), and this is sharp.