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

TL;DR

This paper extends the Lax Equivalence Theorem to unbounded self-adjoint operators, establishing conditions for the stability of Moore-Penrose inverse approximations and highlighting limitations in solving certain Schrödinger operator equations.

Contribution

It generalizes the Lax Equivalence Theorem to unbounded self-adjoint operators and analyzes the stability of Moore-Penrose inverse approximations in this context.

Findings

Continuity and strong convergence of Moore-Penrose inverse approximations depend on boundedness of their norms.

Arbitrary bounded schemes for non-closed range operators are inherently unstable.

Global approximate solutions are infeasible for certain Schrödinger operators with non-closed range.

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