Accidental Degeneracy of an Elliptic Differential Operator: A Clarification in Terms of Ladder Operators

We consider the linear, second-order elliptic, Schrödinger-type differential operator L:=?12?2+r22. Because of its rotational invariance, that is it does not change under SO(3) transformations, eigenvalue problem ?12?2+r22f(x,y,z)=?f(x,y,z) can be studied more conveniently in spherical polar coordinates. It already known eigenfunctions depend on three parameters. The so-called accidental degeneracy L occurs when eigenvalues one such parameters only. exploited ladder operators to reformulate degeneracy, so as provide a new way describe elliptic PDE problems.