Generalization of Lax equivalence theorem on unbounded self-adjoint operators with applications to Schrödinger operators

Define A a unbounded self-adjoint operator on Hilbert space X. Let {An}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \{ A_n \} $$\end{document} be its resolvent approximation sequence with closed range R(An)(n∈N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal {R}}(A_n) (n \in \mathrm {N}) $$\end{document}, that is, An(n∈N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ A_n (n \in \mathrm {N}) $$\end{document} are all self-adjoint on Hilbert space X and s-limn→∞Rλ(An)=Rλ(A)(λ∈C\R),whereRλ(A):=(λI-A)-1.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathop {s-\lim }\limits _{n \rightarrow \infty } R_\lambda (A_n) = R_\lambda (A)\quad (\lambda \in \mathrm {C} \setminus \mathrm {R}), \ \text {where} \ R_ \lambda (A) := (\lambda I-A)^{-1}. \end{aligned}$$\end{document}The Moore–Penrose inverse An†∈B(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ A^\dagger _n \in {\mathcal {B}}(X) $$\end{document} is a natural approximation to the Moore–Penrose inverse A†\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ A^\dagger $$\end{document}. This paper shows that: A†\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ A^\dagger $$\end{document} is continuous and strongly converged by {An†}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \{ A^\dagger _n \} $$\end{document} if and only if supn‖An†‖<+∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sup \nolimits _n \Vert A^\dagger _n \Vert < +\infty $$\end{document}. On the other hand, this result tells that arbitrary bounded computational scheme {An†}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \{ A^\dagger _n \} $$\end{document} induced by resolvent approximation {An}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \{ A_n \} $$\end{document} is naturally instable (that is, supn‖An†‖=∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sup _n \Vert A^\dagger _n \Vert = \infty $$\end{document}) for any self-adjoint operator equation with non-closed range, for example, free Schrödinger operator, Schrödinger operator with Coulomb potential and Schrödinger operator in model of many particles. This implies the infeasibility to globally and approximately solve non-closed range self-adjoint operator equation by resolvent approximation.