Generalization of Lax Equivalence Theorem on Unbounded Self-adjoint Operators with Applications to Schrödinger Operators

Abstract

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{$\genfrac{}{}{0ex}{}{s-\lim }{n\to \infty }$}} R_λ(A_n) = R_λ(A)\quad (λ\in \mathrm{C} \setminus \mathrm{R}), \ \textrm{where} \ R_ λ(A) := (λI-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 $\underset{n}{\sup }\Vert A_n^{†}\Vert <+\infty$. On the other hand, this result tells that arbitrary bounded computational scheme $\{A_n^{†}\}$ induced by resolvent approximation $\{A_n\}$ is naturally instable (that is, ${\sup }_n\Vert A_n^{†}\Vert =\infty$) for any self-adjoint operator equation with non-closed range, for example, free Schrödinger operator, Schrödinger operator with Coulumb potential and Schrödinger 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.

arXiv commentNinth Version

References in corpus (1)

• On the two mutually independent factors that determine the convergence of least-squares projection method