For the Hilbert space of a single mode in quantum optics (i.e. the Fock space), I'd like to be able to capture the intuition that eigenstates of the operators $a_g=ga+\sqrt{g^2-1}a^\dagger$ look more and more like position eigenstates (with eigenvalue going to zero) as $g\rightarrow\infty$.
Defining a sequence $T_n=g^{-1}\left(ga+\sqrt{g^2-1}a^\dagger\right)$, I'd like to say that it converges in some sense to $T=a+a^\dagger$. Obviously since $T_n-T$ is unbounded, looking at norms won't be useful. But for coherent states $\left\vert\alpha\right\rangle$, we have:
\begin{equation} \begin{aligned} \|(T_n-T)\left\vert\alpha\right\rangle\|&=\|[g^{-1}\left(ga+\sqrt{g^2-1}a^\dagger\right)-(a+a^\dagger)]\left\vert\alpha\right\rangle\| \\ &=\left(\sqrt{1-1/g^2}-1\right)\|a^\dagger\left\vert\alpha\right\rangle\| \\ &=\left(\sqrt{1-1/g^2}-1\right)\sqrt{\left\langle\alpha\right\vert aa^\dagger\left\vert\alpha\right\rangle} \\ &=\left(\sqrt{1-1/g^2}-1\right)\sqrt{\left\vert\alpha\right\vert^2+1}. \end{aligned} \end{equation} Then we have \begin{equation} \lim_{g\rightarrow\infty}\|(T_n-T)\left\vert\alpha\right\rangle\|=0\hspace{10pt}\forall\alpha\in\mathbb{C}. \end{equation} This looks sort of like strong operator convergence, since the coherent states are a basis for the Fock space and the argument works for $\textit{finite}$ linear combinations of coherent states, but of course there are vectors in the Fock space that are not in the domain of $T_n-T$, which is already only defined on the intersection of the domains of $T_n$ and $T$.
I'm guessing that these issues are why I can't find much about "convergence" of sequences of unbounded operators, but is there something that captures this idea?