**1. Introduction**

In this paper, we intend to treat an elliptic PDE (Among the numerous textbooks on elliptic PDEs, we think that Gilbarg and Trudinger's book [1], first published in 1998 and then again in 2001 and 2015, is the main contribution to acquire the necessary knowledge on this fascinating topic. On the other hand, the main notions to tackle the typical mathematical physics problems can be found in [2], for example.) with a special focus on the property of the *degeneracy* of its spectrum.

To begin with, we consider the following elliptic PDE:

$$
\left[ -\frac{1}{2}\nabla^2 + \frac{r^2}{2} \right] f(x, y, z) = \lambda f(x, y, z),
\tag{1}
$$

where *r* = *x*<sup>2</sup> + *y*<sup>2</sup> + *z*2, and the function *f*(*x*, *y*, *z*) belongs to the following Hilbert space:

$$\mathcal{H} = \left\{ f(\cdot) \in L^2(\mathbb{R}^3) \cap \mathbb{C}^2(\mathbb{R}^3) \mid \lim\_{r \to \infty} f(x, y, z) = 0 \right\}. \tag{2}$$

As is known, <sup>∇</sup><sup>2</sup> is the Laplacian operator:

$$
\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}.
$$

The operator <sup>L</sup> :<sup>=</sup> <sup>−</sup><sup>1</sup> 2 <sup>∇</sup><sup>2</sup> <sup>+</sup> *<sup>r</sup>*<sup>2</sup> <sup>2</sup> satisfies the property of rotational invariance, i.e., it is invariant under *SO*(3) transformations. Addressing the problem (1) in polar coordinates is not difficult, and it is well known that the eigenfuctions in H depend on three parameters, say, *l*, *m*, *n*, whereas the eigenvalues only depend on *n*, meaning that L is a degenerate operator. However, there are different kinds of degeneracy: If the eigenvalues *λ<sup>i</sup>* are independent of *m*, that is called *natural degeneracy*. If *λ<sup>i</sup>* are independent of *l*, *accidental*

**Citation:** De Marchis, R.; Palestini, A.; Patrì, S. Accidental Degeneracy of an Elliptic Differential Operator: A Clarification in Terms of Ladder Operators. *Mathematics* **2021**, *9*, 3005. https://doi.org/10.3390/ math9233005

Academic Editor: Andrea Scapellato

Received: 26 October 2021 Accepted: 22 November 2021 Published: 23 November 2021

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*degeneracy* occurs. Namely, we focus on accidental degeneracy and on its relationship with ladder operators (a similar procedure applied to spherical hydrogen atom eigenfuctions can be found in [3]).

Recent papers in which the various types of degeneracy are treated are [4–6], just to cite a few.

The paper is organized as follows: In Section 2 the main notions and a selection of useful results on invariance and degeneracy are presented. In Section 3, the ladder operators are introduced and summarized. Section 4 intends to describe the accidental degeneracy of the operator L in detail. Section 5 features a final discussion and the possible future developments of this theory.
