Suppose that for a sequence $\{A_n\}_n$ of bounded self-adjoint operators in a Hilbert space $\mathcal H$ we have $e^{itA_n} \to e^{itA}$ strongly, for all $t \in \mathbb R$, where $A$ is a (possibly unbounded) self-adjoint operator.
If we have a sequence $\{t_n\}_n$ of real numbers such that $t_n \to t$, can we conclude that $e^{it_n A_n} \to e^{it A}$ strongly?
Equivalently, can we show that for every $f \in \mathcal H$ $$ \|e^{it_n A_n}f-e^{it A_n}f\|^2 = \int_{\mathbb R} |e^{it_n\lambda}-e^{it\lambda}|^2 d\mu_f^{A_n}(\lambda) \to 0 \quad ?$$ ($\mu_f^{A_n}$ is the finite measure taking a Borel set $\Delta$ to $\mu_f^{A_n}(\Delta) := \|E_n(\Delta)f\|^2$, where $E_n$ is the spectral measure of $A_n$.)