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

**Roberto De Marchis †, Arsen Palestini \*,† and Stefano Patrì †**

MEMOTEF, Faculty of Economics, Sapienza University of Rome, Via del Castro Laurenziano 9, 00161 Rome, Italy; Roberto.Demarchis@uniroma1.it (R.D.M.); Stefano.Patri@uniroma1.it (S.P.)

**\*** Correspondence: Arsen.Palestini@uniroma1.it

† These authors contributed equally to this work.

**Abstract:** We consider the linear, second-order elliptic, Schrödinger-type differential operator L := −1 2 <sup>∇</sup><sup>2</sup> <sup>+</sup> *<sup>r</sup>*<sup>2</sup> <sup>2</sup> . Because of its rotational invariance, that is it does not change under *SO*(3) transformations, the eigenvalue problem ' −1 2 <sup>∇</sup><sup>2</sup> <sup>+</sup> *<sup>r</sup>*<sup>2</sup> 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.

**Keywords:** degeneracy; elliptic PDE; ladder operator; commuting operator; eigenvalues
