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 : =− (∇^2)/2 +r^2/2 . Because of its rotational invariance, that is it does not change under SO ( 3 ) transformations, the eigenvalue problem [− (∇^2)/2 +r^2/2]f ( x, y, z ) = λ f ( x, y, z ) can be studied more conveniently in spherical polar coordinates. It is already known that the eigenfunctions of the problem depend on three parameters. The so-called accidental degeneracy of L occurs when the eigenvalues of the problem depend on one of such parameters only. We exploited ladder operators to reformulate accidental degeneracy, so as to provide a new way to describe degeneracy in elliptic PDE problems.