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.

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无界自伴随算子上的 Lax 等价定理的推广，并应用于 Schrödinger 算子

在希尔伯特空间 $X$ 上定义一个无界自伴随算子 $A$。设$ \{ A_n \} $ 为其具有闭区间的解析逼近序列$ \mathcal{R}(A_n) (n \in \mathrm{N}) $，即$ A_n (n \in \mathrm{N }) $ 在希尔伯特空间 $ X $ 和 \begin{equation*} \hbox{ \raise-2mm\hbox{$\textstyle s-\lim \atop \scriptstyle {n \to \infty}$ 上都是自伴随的}} R_\lambda (A_n) = R_\lambda (A)\quad (\lambda \in \mathrm{C} \setminus \mathrm{R}), \ \textrm{where} \ R_ \lambda(A) ： = (λ IA)^{-1}。\end{equation*} Moore-Penrose 逆 $ A^\dagger_n \in \mathcal{B}(X) $ 是 Moore-Penrose 逆 $ A^\dagger $ 的自然近似。本文表明： $ A^\dagger $ 是连续的，并且被 $ \{ A^\dagger_n \} $ 强收敛当且仅当 $ \sup\limits_n \Vert A^\dagger_n \Vert < +\infty $。另一方面，