**4. Accidental Degeneracy of the Spectrum of** L<sup>1</sup>

Here, we illustrate the main result, which is absent in the literature so far, to the best of our knowledge. We intend to determine the suitable ladder operators for the degeneracy with respect to the parameter -, which is denoted as accidental degeneracy.

The so-called *accidental degeneracy* of the spectrum of the operator L<sup>1</sup> consists of the independence of the eigenvalues *λ<sup>n</sup>* from the parameter -. We explain also this type of degeneracy with the help of ladder operators, denoted by T(±) <sup>2</sup> . Such ladder operators map an eigenfunction *ψn*,-,*m*(**r**) associated with the eigenvalue *λ<sup>n</sup>* either to the null function or to another eigenfunction, denoted by:

$$
\psi\_{n,\ell',m'}(\mathbf{r}) = \mathbb{T}\_2^{(\pm)} \psi\_{n,\ell,m}(\mathbf{r}) .
$$

The two eigenfunctions belong to the same eigenspace of *λn*, that is the value of *n* is the same in both of them, whereas the two values of are different, and the two values of *m* may be either equal or different.

First of all, we establish the conditions for the functions *ψn*,-,*m*(**r**) and <sup>T</sup>(±) <sup>2</sup> *ψn*,-,*m*(**r**) to be eigenfunctions of L<sup>1</sup> associated with the same eigenvalue *<sup>λ</sup>n*. Namely, by virtue of Theorem 1, any ladder operator T(±) <sup>2</sup> has to satisfy the following equality:

$$[\mathcal{Z}, \mathbb{T}\_2^{(\pm)}] = 0.\tag{14}$$

If *g*(*r*) is any function depending on the polar coordinate *r* only, and the operator:

$$
\widetilde{\mathcal{L}}\_{\mathcal{S}} := -\frac{1}{2}\nabla^2 + \mathcal{g}(r),
$$

is defined on the Hilbert space (8), it follows that every operator L1*<sup>g</sup>* is endowed with rotational invariance. Moreover, there are only two particular circumstances where the operator L1*<sup>g</sup>* ≡ L<sup>1</sup> has a further invariance, which is then "purely accidental" and is responsible for accidental degeneracy. Such cases occur if either *<sup>g</sup>*(*r*) = *<sup>r</sup>*<sup>2</sup> <sup>2</sup> or *<sup>g</sup>*(*r*) = <sup>−</sup><sup>1</sup> *r* . The latter case was extensively treated in [3], so we focus on the former case.

A synthetic explanation may sound as follows: We know that the eigenfunction *ψn*,-,*m*(**r**) is associated with the eigenvalues -(- + 1), with respect to which *ψn*,-,*m*(**r**) is also an eigenfunction of the operator *A*(*θ*, *ϕ*) in (7). Furthermore, since the variation of - between two consecutive values is two, this implies that the eigenvalue of *A*(*θ*, *ϕ*), which is subsequent after -(- + 1), is (- + 2)(- + 3). Hence, the raising operator T(+) <sup>2</sup> , whose expression is to be identified, must induce the shift (- + 2)(- + 3) − -(- + 1) = 4- + 6 on the eigenvalues of *A*(*θ*, *ϕ*).

In order to do that, by virtue of Theorem 5, the raising operator T(+) <sup>2</sup> has to satisfy the following condition:

$$[A(\theta,\varphi),\mathbb{T}\_2^{(+)}]\psi\_{n,\overline{l},\mathfrak{M}}(\mathbf{r})=(4\ell+6)\left[\mathbb{T}\_2^{(+)}\psi\_{n,\overline{l},\mathfrak{M}}(\mathbf{r})\right],\tag{15}$$

where *ψn*,¯ -,*m*¯ (*r*) is a particular eigenfunction of *A*(*θ*, *ϕ*). Therefore, we are supposed to identify an operator that verifies both conditions (14) and (15). The underlying degeneracies have different natures. On the one hand, *natural degeneracy* is clarified by the ladder operators T(±) <sup>1</sup> given in (13) and obtained as combinations of the angular momentum operators, and this is due to the fact that the ladder operators have to induce a shift of one unit on the parameter *m*. On the other hand, *accidental degeneracy* has to be clarified by operators T(+) <sup>2</sup> , which are obtained from the combinations of the components of a tensor, because such operators have to induce a shift of two units on the parameter -.

The invariance of the operator L is illustrated by the next result.

**Theorem 7.** *All the components of the following second-rank tensor:*

$$T\_{i\bar{j}} = -\frac{\partial}{\partial r\_i} \frac{\partial}{\partial r\_{\bar{j}}} + r\_i r\_{\bar{j}\prime} \qquad \qquad \text{for } i, j = 1, 2, 3,\tag{16}$$

*where r*1, *r*2, *r*<sup>3</sup> *are the coordinates of* **r***, satisfy the commutation identity* [L, *Tij*] = 0*, i.e.,* L *is invariant under the action of all components.*

**Proof.** We can employ the following property of the commutator, which holds for all *A*, *B*, and *C*:

$$[AB, \ C] = A[B, \ C] + [A, \ C]B.$$

Expanding the quantity [L, *Tij*] yields (Some calculations are omitted for the sake of brevity. However, all the calculations are available upon request to the authors.):

$$[\mathcal{L}, T\_{ij}] = \left[ -\frac{1}{2}\nabla^2 + \frac{r^2}{2}, -\frac{\partial}{\partial r\_i}\frac{\partial}{\partial r\_j} + r\_i r\_j \right] = \left[ -\frac{1}{2}\nabla^2, r\_i r\_j \right] + \left[ \frac{r^2}{2}, -\frac{\partial}{\partial r\_i}\frac{\partial}{\partial r\_j} \right]$$

$$= -\frac{1}{2}\sum\_{k=1}^3 \left\{ \left[ -\frac{1}{2}\nabla^2, r\_i r\_j \right] + \left[ r\_{k'}^2 - \frac{\partial}{\partial r\_i}\frac{\partial}{\partial r\_j} \right] \right\}$$

$$= -\frac{1}{2}\left( \frac{\partial}{\partial r\_j} r\_i + \frac{\partial}{\partial r\_i} r\_j + r\_i \frac{\partial}{\partial r\_j} + r\_j \frac{\partial}{\partial r\_i} - r\_j \frac{\partial}{\partial r\_i} - r\_i \frac{\partial}{\partial r\_j} - \frac{\partial}{\partial r\_i} r\_j - \frac{\partial}{\partial r\_j} r\_i \right) = 0,$$

meaning that L is invariant under the action of all nine components *Tij*.

The components *Tij* are the further linear operators that accidentally commute with L, in addition to M1, M<sup>2</sup> and M3.

Given the above-mentioned *Tij*, we can consider the following operators:

$$
\mathcal{T}\_1 = T\_{12}, \quad \qquad \qquad \mathcal{T}\_2 = \frac{T\_{22} - T\_{11}}{2},
$$

so that we are able to define the following ladder operators:

$$\mathbb{T}\_2^{(\pm)} := \mathcal{T}\_1 \pm i \mathcal{T}\_2 = -\frac{\partial}{\partial x} \frac{\partial}{\partial y} + xy \pm \frac{i}{2} \left( \frac{\partial^2}{\partial x^2} - \frac{\partial^2}{\partial y^2} + y^2 - x^2 \right) \mathcal{T}\_1$$

where *<sup>i</sup>* <sup>=</sup> √−1.

**Theorem 8.** *The ladder operators* T(±) <sup>2</sup> *satisfy the following commutation identity:*

$$[\mathbb{M}\_3, \mathbb{T}\_2^{(\pm)}] = \pm 2\mathbb{T}\_2^{(\pm)}.\tag{17a}$$

**Proof.** If we expand the expression of the commutator in the left-hand side of (17a), we obtain:

$$\begin{split} \left[\mathbb{M}\_{3},\mathbb{T}\_{2}^{(\pm)}\right] &= i \left[y\frac{\partial}{\partial x}-\text{x},\frac{\partial}{\partial y},-\frac{\partial}{\partial x}\frac{\partial}{\partial y}+\text{xy}\pm\frac{i}{2}\left(\frac{\partial^{2}}{\partial x^{2}}-\frac{\partial^{2}}{\partial y^{2}}+y^{2}-x^{2}\right)\right] \\ &= -i \left[y\frac{\partial}{\partial x},\frac{\partial}{\partial x}\frac{\partial}{\partial y}\right]+i \left[y\frac{\partial}{\partial x},\text{xy}\right]+i \left[\text{x}\frac{\partial}{\partial y},\frac{\partial}{\partial x}\frac{\partial}{\partial y}\right]-i \left[\text{x}\frac{\partial}{\partial y},\text{xy}\right] \\ &\pm\frac{1}{2}\left\{\left[y\frac{\partial}{\partial x},\frac{\partial}{\partial y}\frac{\partial}{\partial y}\right]+\left[y\frac{\partial}{\partial x},\text{x}^{2}\right]+\left[\text{x}\frac{\partial}{\partial y},\frac{\partial}{\partial x}\frac{\partial}{\partial x}\right]+\left[\text{x}\frac{\partial}{\partial y},\text{y}^{2}\right]\right\} \\ &= i\left(\frac{\partial^{2}}{\partial x^{2}}-\frac{\partial^{2}}{\partial y^{2}}+y^{2}-x^{2}\right)\pm\left(-2\frac{\partial}{\partial x}\frac{\partial}{\partial y}+2xy\right)=\pm 2\text{T}\_{2}^{(\pm)}. \end{split}$$

Using Relations (12a), (12b), (14), and (17a), we can establish the following actions of the operators T(±) <sup>2</sup> on the eigenfunctions *ψn*,-,*m*(**r**):

$$\mathbb{T}\_2^{(+)}\psi\_{n,\ell,m}(\mathbf{r}) = \sum\_{k=0}^{(n-m-2)/2} c\_k \psi\_{n,n-2k,m+2}(\mathbf{r}),\\\mathbb{T}\_2^{(-)}\psi\_{n,\ell,m}(\mathbf{r}) = \sum\_{k=0}^{(n-m+2)/2} c\_k \psi\_{n,n-2k,m-2}(\mathbf{r}).$$

In order to prove that T(±) <sup>2</sup> are the ladder operators that give a clarification of accidental degeneracy, we have to determine the commutators [*A*(*θ*, *ϕ*),T(+) <sup>2</sup> ] and [*A*(*θ*, *<sup>ϕ</sup>*),T(−) <sup>2</sup> ]. As in Theorem 8, we also obtain the commutators:

[M1, <sup>T</sup>(+) <sup>2</sup> ] = *i* ' *z ∂ <sup>∂</sup><sup>y</sup>* <sup>−</sup> *<sup>y</sup> ∂ <sup>∂</sup><sup>z</sup>* , <sup>−</sup> *<sup>∂</sup> ∂x ∂ ∂y* + *xy* + *i* 2 - *∂*2 *<sup>∂</sup>x*<sup>2</sup> <sup>−</sup> *<sup>∂</sup>*<sup>2</sup> *<sup>∂</sup>y*<sup>2</sup> <sup>+</sup> *<sup>y</sup>*<sup>2</sup> <sup>−</sup> *<sup>x</sup>*<sup>2</sup> ( = *i* ' *z ∂ <sup>∂</sup><sup>y</sup>* , *xy*( + *i* ' *y ∂ <sup>∂</sup><sup>z</sup>* , *<sup>∂</sup> ∂x ∂ ∂y* ( − 1 2 ' *z ∂ <sup>∂</sup><sup>y</sup> <sup>y</sup>*<sup>2</sup> ( − 1 2 ' *y ∂ <sup>∂</sup><sup>z</sup>* , *<sup>∂</sup>*<sup>2</sup> *∂y*<sup>2</sup> ( <sup>=</sup> *<sup>∂</sup> ∂y ∂ <sup>∂</sup><sup>z</sup>* <sup>−</sup> *yz* <sup>+</sup> *ixz* <sup>−</sup> *<sup>i</sup> <sup>∂</sup> ∂x ∂ ∂z* , [M2, <sup>T</sup>(+) <sup>2</sup> ] = *i* ' *x ∂ <sup>∂</sup><sup>z</sup>* <sup>−</sup> *<sup>z</sup> <sup>∂</sup> <sup>∂</sup><sup>x</sup>* , <sup>−</sup> *<sup>∂</sup> ∂x ∂ ∂y* + *xy* + *i* 2 - *∂*2 *<sup>∂</sup>x*<sup>2</sup> <sup>−</sup> *<sup>∂</sup>*<sup>2</sup> *<sup>∂</sup>y*<sup>2</sup> <sup>+</sup> *<sup>y</sup>*<sup>2</sup> <sup>−</sup> *<sup>x</sup>*<sup>2</sup> ( = *i* ' *∂ ∂x ∂ <sup>∂</sup><sup>y</sup>* , *<sup>x</sup> <sup>∂</sup> ∂z* ( + *i* ' *xy*, *<sup>z</sup> <sup>∂</sup> ∂x* ( + 1 2 ' *∂*2 *<sup>∂</sup>x*<sup>2</sup> , *<sup>x</sup> <sup>∂</sup> ∂z* ( + 1 2 ' *<sup>x</sup>*2, *<sup>z</sup> <sup>∂</sup> ∂x* ( <sup>=</sup> *<sup>∂</sup> ∂x ∂ <sup>∂</sup><sup>z</sup>* <sup>−</sup> *xz* <sup>−</sup> *iyz* <sup>+</sup> *<sup>i</sup> <sup>∂</sup> ∂y ∂ ∂z* ,

from which we can prove the following fundamental result.

**Theorem 9** (**Theorem of accidental degeneracy**)**.** *The commutator A*(*θ*, *ϕ*),T(+) 2 *is the operator:*

$$\mathbb{E}\left[A(\theta,\varphi),\mathbb{T}\_2^{(+)}\right] = 4\mathbb{T}\_2^{(+)}\mathbb{M}\_3 + 6\mathbb{T}\_2^{(+)} + \left(-2i\frac{\partial}{\partial x}\frac{\partial}{\partial z} + 2\frac{\partial}{\partial y}\frac{\partial}{\partial z} + 2ixz - 2yz\right)\mathbb{T}\_1^{(+)} \mathbb{M}\_3$$

*where* T(+) <sup>1</sup> *is the raising operator of the natural degeneracy in* (13)*.*

**Proof.** Using the relation *<sup>A</sup>*(*θ*, *<sup>ϕ</sup>*) <sup>≡</sup> <sup>M</sup><sup>2</sup> <sup>1</sup> <sup>+</sup> <sup>M</sup><sup>2</sup> <sup>2</sup> <sup>+</sup> <sup>M</sup><sup>2</sup> <sup>3</sup> and expanding the left-hand side, we have:

 *A*(*θ*, *ϕ*), T(+) 2 = M2 <sup>1</sup> <sup>+</sup> <sup>M</sup><sup>2</sup> <sup>2</sup> <sup>+</sup> <sup>M</sup><sup>2</sup> 3, <sup>T</sup>(+) 2 = M<sup>1</sup> <sup>M</sup>1, <sup>T</sup>(+) 2 + <sup>M</sup>1, <sup>T</sup>(+) 2 M<sup>1</sup> + M<sup>2</sup> <sup>M</sup>2, <sup>T</sup>(+) 2 + <sup>M</sup>2, <sup>T</sup>(+) 2 M2 +M<sup>3</sup> <sup>M</sup>3, <sup>T</sup>(+) 2 + <sup>M</sup>3, <sup>T</sup>(+) 2 M3 = *i* - *z ∂ <sup>∂</sup><sup>y</sup>* <sup>−</sup> *<sup>y</sup> ∂ ∂z* - *∂ ∂y ∂ <sup>∂</sup><sup>z</sup>* <sup>−</sup> *yz* <sup>+</sup> *ixz* <sup>−</sup> *<sup>i</sup> <sup>∂</sup> ∂x ∂ ∂z* + *i* - *∂ ∂y ∂ <sup>∂</sup><sup>z</sup>* <sup>−</sup> *yz* <sup>+</sup> *ixz* <sup>−</sup> *<sup>i</sup> <sup>∂</sup> ∂x ∂ ∂z z ∂ <sup>∂</sup><sup>y</sup>* <sup>−</sup> *<sup>y</sup> ∂ ∂z* + *i* - *x ∂ <sup>∂</sup><sup>z</sup>* <sup>−</sup> *<sup>z</sup> <sup>∂</sup> ∂x* - *∂ ∂x ∂ <sup>∂</sup><sup>z</sup>* <sup>−</sup> *xz* <sup>−</sup> *iyz* <sup>+</sup> *<sup>i</sup> <sup>∂</sup> ∂y ∂ ∂z* + *i* - *∂ ∂x ∂ <sup>∂</sup><sup>z</sup>* <sup>−</sup> *xz* <sup>−</sup> *iyz* <sup>+</sup> *<sup>i</sup> <sup>∂</sup> ∂y ∂ ∂z x ∂ <sup>∂</sup><sup>z</sup>* <sup>−</sup> *<sup>z</sup> <sup>∂</sup> ∂x* + 2*i* - *y ∂ <sup>∂</sup><sup>x</sup>* <sup>−</sup> *<sup>x</sup> <sup>∂</sup> ∂y* '<sup>−</sup> *<sup>∂</sup> ∂x ∂ ∂y* + *xy* + *i* 2 - *∂*2 *<sup>∂</sup>x*<sup>2</sup> <sup>−</sup> *<sup>∂</sup>*<sup>2</sup> *<sup>∂</sup>y*<sup>2</sup> <sup>+</sup> *<sup>y</sup>*<sup>2</sup> <sup>−</sup> *<sup>x</sup>*<sup>2</sup> ( + 2*i* ' − *∂ ∂x ∂ ∂y* + *xy* + *i* 2 - *∂*2 *<sup>∂</sup>x*<sup>2</sup> <sup>−</sup> *<sup>∂</sup>*<sup>2</sup> *<sup>∂</sup>y*<sup>2</sup> <sup>+</sup> *<sup>y</sup>*<sup>2</sup> <sup>−</sup> *<sup>x</sup>*<sup>2</sup> (*y ∂ <sup>∂</sup><sup>x</sup>* <sup>−</sup> *<sup>x</sup> <sup>∂</sup> ∂y* .

Expanding the right-hand side yields:

$$4\mathbb{T}\_2^{(+)}\mathbb{M}\_3 + 6\mathbb{T}\_2^{(+)} + \left(-2i\frac{\partial}{\partial x}\frac{\partial}{\partial z} + 2\frac{\partial}{\partial y}\frac{\partial}{\partial z} + 2ixz - 2yz\right)\mathbb{T}\_1^{(+)}$$

$$= 4i\left(-\frac{\partial}{\partial x}\frac{\partial}{\partial y} + xy + \frac{i}{2}\frac{\partial^2}{\partial x^2} - \frac{i}{2}\frac{\partial^2}{\partial y^2} + \frac{iy^2}{2} - \frac{ix^2}{2}\right)\left(y\frac{\partial}{\partial x} - x\frac{\partial}{\partial y}\right)$$

$$+ 6\left(-\frac{\partial}{\partial x}\frac{\partial}{\partial y} + xy + \frac{i}{2}\frac{\partial^2}{\partial x^2} - \frac{i}{2}\frac{\partial^2}{\partial y^2} + \frac{iy^2}{2} - \frac{ix^2}{2}\right)$$

$$+ \left(-2i\frac{\partial}{\partial x}\frac{\partial}{\partial z} + 2\frac{\partial}{\partial y}\frac{\partial}{\partial z} + 2ixz - 2yz\right)\left(iz\frac{\partial}{\partial y} - iy\frac{\partial}{\partial z} - x\frac{\partial}{\partial z} + z\frac{\partial}{\partial z}\right)$$

With the help of some algebra, we can recognize that the two expansions are equal; hence, the proof is complete.

Theorem <sup>9</sup> and the identity T(+) <sup>1</sup> *ψn*-- = 0 lead to establishing the action of the commutator [*A*(*θ*, *ϕ*), T(+) <sup>2</sup> ] on the eigenfunction *ψn*,-,-, obtained when positing *m* = -.

$$\left[A(\theta,\eta)\_{\prime}\operatorname{\mathsf{T}}\_{2}^{(+)}\right]\psi\_{n,\ell,\ell}$$

$$=\left[4\operatorname{\mathsf{T}}\_{2}^{(+)}\operatorname{\mathsf{M}}\_{3} + 6\operatorname{\mathsf{T}}\_{2}^{(+)} + \left(-2i\frac{\partial}{\partial x}\frac{\partial}{\partial z} + 2\frac{\partial}{\partial y}\frac{\partial}{\partial z} + 2ix\operatorname{\mathsf{z}} - 2yz\right)\operatorname{\mathsf{T}}\_{1}^{(+)}\right]\psi\_{n,\ell,\ell}$$

$$=\left[4\operatorname{\mathsf{T}}\_{2}^{(+)}\operatorname{\mathsf{M}}\_{3} + 6\operatorname{\mathsf{T}}\_{2}^{(+)}\right]\psi\_{n,\ell,\ell} + \left(-2i\frac{\partial}{\partial x}\frac{\partial}{\partial z} + 2\frac{\partial}{\partial y}\frac{\partial}{\partial z} + 2ix\operatorname{\mathsf{z}} - 2yz\right)\left[\operatorname{\mathsf{T}}\_{1}^{(+)}\operatorname{\mathsf{V}}\_{n,\ell,\ell}\right]\psi\_{n,\ell,\ell}$$

$$=\left(4\ell + 6\right)\left[\operatorname{\mathsf{T}}\_{2}^{(+)}\psi\_{n,\ell,\ell}\right]\_{\prime}$$

that is we have found the fundamental commutator:

$$\mathbb{E}\left[A(\theta,\varphi),\mathbb{T}\_2^{(+)}\right]\psi\_{n,\ell,\ell} = (4\ell+6)\left[\mathbb{T}\_2^{(+)}\psi\_{n,\ell,\ell}\right].\tag{17b}$$

.

The function T(+) <sup>2</sup> *ψn*,-, is either the null function or a simultaneous eigenfunction of the operator L, with respect to the same eigenvalue *λ<sup>n</sup>* as *ψn*,-,-, and of the operators *A*(*θ*, *ϕ*), M3, with respect to the eigenvalues -(- + 1) + 4- + 6 ≡ (- + 2)(- + 3) and - + 2, respectively, that is:

$$
\mathbb{T}\_2^{(+)} \psi\_{n,\ell,\ell} = \mathbb{C}\_{\ell+2} \,\psi\_{n,\ell+2,\ell+2} \cdot \,\,\,\,
$$

Furthermore, the raising operator T(+) <sup>2</sup> provides a clarification of the accidental degeneracy of the spectrum of the operator L<sup>1</sup> because its iterated action on the eigenfunction *ψn*,0,0, where *n* is even, gives:

$$\mathbb{T}\_2^{(+)}\psi\_{n,0,0} = \mathbb{C}\_2\psi\_{n,2,2}, \quad \mathbb{T}\_2^{(+)}\psi\_{n,2,2} = \mathbb{C}\_4\psi\_{n,4,4}, \qquad \dots, \quad \mathbb{T}\_2^{(+)}\psi\_{n,n-2,n-2} = \mathbb{C}\_n\psi\_{n,n,n-2}$$

and analogously with *n* odd:

$$\mathbb{T}\_2^{(+)}\psi\_{n,1,1} = \mathbb{C}\_3\,\psi\_{n,3,3}, \quad \mathbb{T}\_2^{(+)}\psi\_{n,3,3} = \mathbb{C}\_5\,\psi\_{n,5,5}, \qquad \dots, \quad \mathbb{T}\_2^{(+)}\psi\_{n,n-2,n-2} = \mathbb{C}\_n\,\psi\_{n,n,n-1}$$

where the coefficients *Ci* are coefficients of normalization, that is the action of the raising operator T(+) <sup>2</sup> on the eigenfunctions *ψn*,-, leaves the parameters *n* unchanged and modifies the parameters -, *m*, only.

Regarding the operator T(−) <sup>2</sup> , we have the relation (it can be proven by means of the same strategy as in Theorem 9):

$$\mathbb{E}\left[A(\theta,\varphi),\mathbb{T}\_2^{(-)}\right] = -4\mathbb{T}\_2^{(-)}\mathbb{M}\_3 + 6\mathbb{T}\_2^{(-)} + \left(-2i\frac{\partial}{\partial x}\frac{\partial}{\partial z} - 2\frac{\partial}{\partial y}\frac{\partial}{\partial z} + 2ixz + 2yz\right)\mathbb{T}\_1^{(-)}\mathbb{M}\_2$$

where T(−) <sup>1</sup> is the lowering operator of the natural degeneracy in (13), from which we obtain the action:

$$\mathbb{E}\left[A(\theta,\,\,\Psi),\,\mathbb{T}\_2^{(-)}\right]\psi\_{n,\ell,-\ell} = (4\ell+6)\left[\mathbb{T}\_2^{(-)}\psi\_{n,\ell,-\ell}\right].\tag{18}$$

Again, for the above reasons, the function T(−) <sup>2</sup> *ψn*,-,− is either the null function or a simultaneous eigenfunction of the operator L, with respect to the same eigenvalue *λ<sup>n</sup>* as *ψn*,-,−-, and of the operators *A*(*θ*, *ϕ*), M3, with respect to the eigenvalues -(- + 1) + 4- + 6 ≡ (- + 2)(- + 3) and −-− 2, respectively, that is:

$$
\mathbb{T}\_2^{(-)}\psi\_{n,\ell,-\ell} = \hat{\mathcal{C}}\_{\ell+2}\psi\_{n,\ell+2,-\ell-2} - 
$$

Since the action of the operator T(−) <sup>2</sup> on the eigenfunctions *ψn*,-,− raises the parameter by two units, as the operator T(+) <sup>2</sup> , we can conclude that there is no lowering operator for the parameter -, but this is not surprising because, by virtue of (17a), the operator T(−) 2 lowers the parameter *m* of the eigenfunctions *ψn*,-,− from − to −- − 2. This means that the parameter cannot change from to -− 2 because otherwise, we would have that:

$$|m| = |-\ell - 2| > \ell - 2,$$

.

which is absurd, due to the constraint |*m*| -

Furthermore, the operator T(−) <sup>2</sup> provides a clarification of the accidental degeneracy of the spectrum of the operator L<sup>1</sup> because its iterated action on the eigenfunction *<sup>ψ</sup>n*,0,0, where *n* is even, gives:

$$\mathbb{T}\_2^{(-)}\psi\_{n,0,0} = \breve{\mathbb{C}}\_2\psi\_{n,2,-2}, \ \ \mathbb{T}\_2^{(-)}\psi\_{n,2,-2} = \breve{\mathbb{C}}\_4\psi\_{n,4,-4}, \ \ \dots, \ \ \mathbb{T}\_2^{(-)}\psi\_{n,n-2,-n+2} = \breve{\mathbb{C}}\_n\psi\_{n,n,-n,-2}$$

and also, if *n* is odd:

$$\mathbb{T}\_2^{(-)}\psi\_{n,1,-1} = \breve{\mathbb{C}}\_3\psi\_{n,3,-3}, \ \mathbb{T}\_2^{(-)}\psi\_{n,3,-3} = \breve{\mathbb{C}}\_5\psi\_{n,5,-5}, \ \dots, \ \mathbb{T}\_2^{(-)}\psi\_{n,n-2,-n+2} = \breve{\mathbb{C}}\_n\psi\_{n,n,-n+2}$$

where the coefficients *<sup>C</sup>*1*<sup>i</sup>* are coefficients of normalization, i.e., the action of the operator T(−) <sup>2</sup> on the eigenfunctions *ψn*,-,− leaves the parameters *n* unchanged and modifies the parameters -, *m*, only.

#### **5. Discussion**

In this paper, we focused on the accidental degeneracy of a second-order, Schrödingertype differential operator, acting on a Hilbert space. Typically, in the theory of PDEs, the concept of degeneracy is connected to the number of parameters on which the eigenvalues depend. Natural degeneracy and accidental degeneracy were reformulated and characterized by using the ladder operators. Such a useful tool can be further employed to provide a new way to describe degeneracy in eigenvalue problems with elliptic operators.

**Author Contributions:** Methodology, S.P.; Writing—original draft, A.P.; Writing—review—editing, R.D.M. All the authors equally contributed to the present work. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study, nor in the writing of the manuscript.
