  Define $ A $ a unbounded self-adjoint operator on Hilbert space $ X $. Let $
\{ A_n \} $ be its resolvent approximation sequence with closed range $
\mathcal{R}(A_n) (n \in \mathrm{N}) $, that is, $ A_n (n \in \mathrm{N}) $ are
all self-adjoint on Hilbert space $ X $ and \begin{equation*} \hbox{
\raise-2mm\hbox{$\textstyle s-\lim \atop \scriptstyle {n \to \infty}$}}
R_\lambda (A_n)
  = R_\lambda (A)\quad (\lambda \in \mathrm{C} \setminus \mathrm{R}), \
\textrm{where} \ R_ \lambda(A) := (\lambda I-A)^{-1}. \end{equation*} The
Moore-Penrose inverse $ A^\dagger_n \in \mathcal{B}(X) $ is a natural
approximation to the Moore-Penrose inverse $ A^\dagger $. This paper shows
that: $ A^\dagger $ is continuous and strongly converged by $ \{ A^\dagger_n \}
$ if and only if $ \sup\limits_n \Vert A^\dagger_n \Vert < +\infty $. On the
other hand, this result tells that arbitrary bounded computational scheme $ \{
A^\dagger_n \} $ induced by resolvent approximation $ \{ A_n \} $ is naturally
instable (that is, $ \sup_n \Vert A^\dagger_n \Vert = \infty $) for any
self-adjoint operator equation with non-closed range, for example, free
Schr\"{o}dinger operator, Schr\"{o}dinger operator with Coulumb potential and
Schr\"{o}dinger operator in model of many particles. This implies the
infeasibility to globally and approximately solve non-closed range self-ajoint
operator equation by resolvent approximation.
