  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 $a=1$.
