We consider the Schrödinger equation i ∂tu+Δu = 0 with initial data in Hs(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 < 1 4 . 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 H1/4(Rn), and this is sharp.

This research has been supported by EPSRC grant EP/E022340/1, ERC grant 277778, and MINECO grants SEV-2011-0087 and MTM2010-16518.

Peer reviewed