Stability of good quantum numbers in ground states

Let $H$ be a self-adjoint operator, bounded from below and let $O$ be a bounded self-adjoint operator with purely discrete spectrum. Suppose that (i) $E(H)=\inf \mathrm{spec}(H)$ is a simple eigenvalue, and (ii) $H$ strongly commutes with $O$. Let $\psi_H$ be the eigenvector associated with $E(H)$. By the assumptions (i) and (ii), $\psi_H$ is an eigenvector of $O$: $O\psi_H=\mu(H)\psi_H$. In the context of quantum mechanics, $\mu(H)$ is called a good quantum number. In this note, we examine the stability of $\mu(H)$ under perturbations of $H$ from a viewpoint of the order theory.