Pointwise Convergence of Solutions to Schrödinger Equations

We study pointwise convergence of the solutions to Schrödinger equations with initial datum f ∈ H(R). The conjecture is that the solution ef converges to f almost everywhere for all f ∈ H(R) if and only if s ≥ 1/4. The conjecture is known true for one spatial dimension and the convergence when s > 1/2 was verified for n ≥ 2. Recently, concrete progresses have been made in R for some s < 1/2. However, when n ≥ 3 no positive result is known for the initial datum f ∈ H(R), s ≤ 1/2. We show that limt→0 ef = f a.e. for f ∈ H(R) whenever s > 1/2− 1/24.