TITLE: Pointwise convergence in functional calculus
QUESTION [5 upvotes]: Let $A_n$ be a family of (bounded) self-adjoint operator converging pointwise to some (unbounded) self-adjoint operator $A,$ i.e. for all $x$ in the domain of $A$
$$\left\lVert A_n x-Ax \right\rVert \rightarrow 0.$$
Does this imply that $e^{it A_n}$ converges pointwise to $e^{itA}$?
I know it holds, if the $A_n$ commute with each other as in this case 
$$\left\lVert T_n(t)x-T_m(t)x \right\rVert \le \int_0^t \left\lVert \frac{d}{ds} T_m(t-s)T_n(s) x \right\rVert \ ds \le t \left\lVert A_n x- A_m x \right\rVert$$
where $T_n =e^{itA_n}.$
Searching the literature myself I noticed that this might be related (especially the frist paragraph of the question):
click me.

REPLY [4 votes]: There are easyer and more direct ways to prove it, but this follows immediatelly as a special case from the Trotter-Kato approximation theorem, see Theorem III.4.8 in 
Engel, Klaus-Jochen; Nagel, Rainer, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics. 194. Berlin: Springer. xxi, 586 p. (2000). ZBL0952.47036.