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

$$\begin{aligned} \mathop {s-\lim }\limits _{n \rightarrow \infty } R_\lambda (A_n) = R_\lambda (A)\quad (\lambda \in \mathrm {C} \setminus \mathrm {R}), \ \text {where} \ R_ \lambda (A) := (\lambda I-A)^{-1}. \end{aligned}$$

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 $\sup \nolimits _n \Vert A^\dagger _n \Vert < +\infty$. 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 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.