I'm learning about the dynamical convergence (i.e, convergence of the unitary group associated with each operator) and resolvent convergence of (unbounded) self-adjoint densely defined operators. I can understand the proof of the initial results, but I can't notice what is the motivation for this. One motivation that I have seen was the following: if $(T_{n})_{n\in\mathbb{N}}$ and $T$ are (unbounded) self-adjoint densely defined operators in a Hilbert space $H$, then the intersection of the domains could be only the null vector, thus the convergence $T_{n}v\to Tv$ will be true for all, possible, $v$, since $v$ is just allowed to be $0$, the convergence is gonna be true. I'm theoretically satisfied with this intuition, but I can't find an example to show that possibility, i.e, a sequence like that with $dom(T)\cap\biggl(\bigcap_{n\in\mathbb{N}}dom(T_{n})\biggr) = \left \{0\right \}$
Just to add some reference on my question, it can be found on "César R. de Oliveira, Intermediate Spectral Theory and Quantum dynamics, Chapter 10".