Consider a family of self-adjoint operators $H_n(\alpha)$ on a Hilbert space $\mathcal{H}$, with $n\in\mathbb{N}$ and $\alpha$ ranging in some bounded real interval $I$, such that, for every $\alpha\in I$, there is a self-adjoint operator $H(\alpha)$ such that $H_n(\alpha)$ converges in the dynamical sense to $H(\alpha)$, i.e., for all $t\in\mathbb{R}$ and $\psi\in\mathcal{H}$ \begin{equation*} \|(e^{-itH_n(\alpha)}-e^{-itH(\alpha)})\psi\|\to0\qquad(n\to\infty). \end{equation*} Now let us consider a smooth evolution $\alpha(t)$ of the parameter $\alpha$ in some bounded real interval $[0,T]$, such that the time-dependent Schrödinger equation is solvable for both $H_n(\alpha(t))$ and $H(\alpha(t))$, i.e. there exist two unitary propagators $U_n(t,t')$, $U(t,t')$ respectively for $H_n(\alpha(t))$ and $H(\alpha(t))$.
My question is: does the property \begin{equation*} \|(U_n(t,t')-U(t,t'))\psi\|\to0\qquad(n\to\infty) \end{equation*} also hold for all $t,t'\in\mathbb{R}$ and $\psi\in\mathcal{H}$, under suitable assumptions?
By hypothesis, this is true if $\alpha(t)$ is constant (since, in that case, we simply have $U(t,t')=e^{-i(t-t')H(\alpha)}$ and $U_n(t,t')=e^{-i(t-t')H_n(\alpha)}$. I was thinking about approximating $\alpha(t)$ with a piecewise constant function, applying the previous remark to every interval in which $\alpha(t)$ is constant, and then extending the result to a more general class of functions.