Generalization of Lax Equivalence Theorem on Unbounded Self-adjoint Operators with Applications to Schr\"{o}dinger Operators

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{$\genfrac{}{}{0px}{}{s-\lim }{n\to \mathrm{\infty }}$}} R_\lambda (A_n) = R_\lambda (A)\quad (\lambda \in $\mathrm{C}$ \setminus $\mathrm{R}$),
$\text{where}$ \ R_ \lambda(A) := (\lambda I-A)^{-1}. \end{equation*} The Moore-Penrose inverse $A_n^{†}\in \mathcal{B}(X)$ is a natural approximation to the Moore-Penrose inverse $A^{†}$. This paper shows that: $A^{†}$ is continuous and strongly converged by $\{A_n^{†}\}$ if and only if $\underset{n}{\sup }\mathrm{\parallel }{A}_n^{†}\mathrm{\parallel }<+\mathrm{\infty }$. On the other hand, this result tells that arbitrary bounded computational scheme $ { A^\dagger_n } $inducedbyresolventapproximation$ { A_n } $ is naturally instable (that is, ${\sup }_n\mathrm{\parallel }{A}_n^{†}\mathrm{\parallel }=\mathrm{\infty }$) 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.