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{$\textstyle s-\lim \atop \scriptstyle {n \to \infty}$}} 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.