On pointwise convergence of Schr\"odinger means

For functions in the Sobolev space $H^s$ and decreasing sequences $t_n\to 0$ we examine convergence almost everywhere of the generalized Schr\"odinger means on the real line, given by $$S^af(x,t_n)=\exp( it_n (-\partial_{xx})^{a/2})f(x);$$ here $a>0$, $a\neq 1$. For decreasing convex sequences we obtain a simple characterization of convergence a.e. for all functions in $H^s$ when $0<s<\min\{a/4,1/4\}$ and $a\neq 1$. We prove sharp quantitative local and global estimates for the associated maximal functions. We also obtain sharp results for the case $a=1$.