This problem if from Methods of modern mathematical physics I :Functional Analysis, by Reed and Simon:
Problem 21: Let $\{A_n\}$ be a sequence of selfadjoint operators on a Hilbert space $H$, and let $A$ be a selfadjoint (not necessarily bounded) operator on $H$. Show that if $A_n \to A$ in the strong resolvent sense, then $$ \mathrm{e}^{itA_n}x \to \mathrm{e}^{itA}x $$ uniformly for $t$ in any finite interval.
I would be grateful for showing that this convergence is locally uniform in $t$.
In this paper you can find more results about the strong resolvent convergence. See the first 4 pages. http://www2.im.uj.edu.pl/actamath/PDF/34-153-163.pdf