  We study pointwise convergence of the solutions to Schr\"odinger equations
with initial datum $f\in H^s(\mathbb R^n)$. The conjecture is that the solution
$e^{it\Delta}f$ converges to $f$ almost everywhere for all $f\in H^s(\mathbb
R^n)$ if and only if $s\ge 1/4$. The conjecture is known true for one spatial
dimension and the convergence when $s>1/2$ was verified for $n\ge 2$. Recently,
concrete progresses have been made in $\mathbb R^2$ for some $s<1/2$. However,
when $n\ge 3$ no positive result is known for the initial datum $f\in
H^s(\mathbb R^n)$, $s\le 1/2$. We show that $\lim_{t\to 0} e^{it\Delta}f= f$
a.e. for $f\in H^s(\mathbb R^3)$ whenever $s>1/2-1/{24}$.
