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) $E(H)=\inf \mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}(H)$ 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.