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

Define A a unbounded self-adjoint operator on Hilbert space X. Let {An} be its resolvent approximation sequence with closed range R( An)(n. N), that is, An(n. N) are all self-adjoint on Hilbert space X and s - lim n.8 R.( An) = R.( A) (.. C \ R), where R.( A) := (.I - A)-1. The Moore-Penrose inverse A+ n. B( X) is a natural approximation to the MoorePenrose inverse A+. This paper shows that: A+ is continuous and strongly converged by {A+ n} if and only if supn A+ n < +8. On the other hand, this result tells that arbitrary bounded computational scheme {A+ n} induced by resolvent approximation {An} is naturally instable (that is, supn A+ n = 8) for any self-adjoint operator equation with non-closed range, for example, free Schrodinger operator, Schrodinger operator with Coulomb potential and Schrodinger 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.