*Natural Degeneracy of a Spectrum*

The first ladder operator <sup>T</sup><sup>1</sup> of <sup>L</sup><sup>1</sup> is already well known in the literature. Namely, it can be easily reconstructed as a combination of the three operators M1, M2, and M3. To be more precise, we take into account the two combinations of the operators M<sup>1</sup> and M<sup>2</sup> that we express in Cartesian and in spherical polar coordinates as follows:

$$\begin{split} \mathbb{T}\_{1}^{(+)} &:= \mathbb{M}\_{1} + i\mathbb{M}\_{2} = i \left( z \frac{\partial}{\partial y} - y \frac{\partial}{\partial z} \right) - \left( x \frac{\partial}{\partial z} - z \frac{\partial}{\partial x} \right) = e^{i\phi} \left( \frac{\partial}{\partial \theta} + \frac{i \cos \theta}{\sin \theta} \frac{\partial}{\partial \phi} \right), \\ \mathbb{T}\_{1}^{(-)} &:= \mathbb{M}\_{1} - i\mathbb{M}\_{2} = i \left( z \frac{\partial}{\partial y} - y \frac{\partial}{\partial z} \right) + \left( x \frac{\partial}{\partial z} - z \frac{\partial}{\partial x} \right) = e^{-i\phi} \left( \frac{i \cos \theta}{\sin \theta} \frac{\partial}{\partial \phi} - \frac{\partial}{\partial \theta} \right), \end{split} \tag{13}$$

which respectively are the *raising operator* and the *lowering operator*.

Since the two ladder operators T(±) <sup>1</sup> satisfy the conditions:

$$[\tilde{\mathcal{L}}, \mathbb{T}\_1^{(\pm)}] = [A(\theta, \mathfrak{q}), \mathbb{T}\_1^{(\pm)}] = 0, \qquad [\tilde{\mathbb{M}}\_3, \mathbb{T}\_1^{(\pm)}] = \pm \mathbb{T}\_1^{(\pm)},$$

we obtain, according the Equations (12a) and (12b), that the functions T(±) <sup>1</sup> *ψn*,-,*<sup>m</sup>* are eigenfunctions of the operators <sup>L</sup><sup>1</sup> and *<sup>A</sup>*(*θ*, *<sup>ϕ</sup>*) with respect to the same eigenvalues *<sup>λ</sup>n*, -(- + 1), respectively, and eigenfunctions of <sup>M</sup><sup>1</sup> <sup>3</sup> with respect to the shifted eigenvalue *<sup>m</sup>* <sup>±</sup> 1.

The action of the ladder operators on the functions *ψn*,-,*<sup>m</sup>* is described by the next result.

**Theorem 6.** *The functions* T(+) <sup>1</sup> *ψn*,-, *and* T(−) <sup>1</sup> *ψn*,-,−*are identically zero.*

**Proof.** If we expand the function T(+) <sup>1</sup> *ψn*,-,-, we can note that (<sup>1</sup> <sup>−</sup> *<sup>u</sup>*2) is a polynomial having degree 2 in *u*. Indicating with the constant K the 2--th derivative of the function w.r.t. *u*, we obtain the following expression:

$$\mathbf{T}\_{1}^{(+)}\psi\_{n,\ell,\ell} = e^{i\phi} \left(\frac{\partial}{\partial\theta} + \frac{i\cos\theta}{\sin\theta}\frac{\partial}{\partial\phi}\right) \left[e^{i\ell\phi}\sin^{\ell}\theta \left(\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\right)^{2\ell}\sin^{2\ell}\theta\right]$$

$$= e^{i(\ell+1)\phi} \left\{\frac{\partial}{\partial\theta} \left[\sin^{\ell}\theta \left(\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\right)^{2\ell}\sin^{2\ell}\theta\right] - \ell\cos\theta\sin^{\ell-1}\theta \left(\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\right)^{2\ell}\sin^{2\ell}\theta\right\}.$$

Now, if we posit cos *<sup>θ</sup>* <sup>=</sup> *<sup>u</sup>* and sin *<sup>θ</sup>* <sup>=</sup> <sup>√</sup> 1 − *u*2, the latest expression becomes:

$$
\epsilon^{i(\ell+1)\phi} \left\{ -\sqrt{1-u^2} \left[ \frac{d}{du} \left( (1-u^2)^{\ell/2} \frac{d^{2\ell}}{du^{2\ell}} (1-u^2)^{\ell} \right) \right] \right.
$$

$$
$$

$$
=\mathcal{K}\epsilon^{i(\ell+1)\phi} \left\{ -\sqrt{1-u^2} \left[ \frac{d}{du} (1-u^2)\ell/2 \right] - \ell u (1-u^2)^{(\ell-1)/2} \right\} = 0.
$$

The pair of ladder operators T(±) <sup>1</sup> provides a *clarification* of that part of the degeneracy of the spectrum of L1, which is called *natural degeneracy*. As a matter of fact, the operator L1, depending on the Laplacian operator <sup>∇</sup><sup>2</sup> and the norm *<sup>r</sup>* of the vector **<sup>r</sup>**, only, has a natural and intrinsic invariance under rotations belonging to the proper rotation group *SO*(3).

The existence of the ladder operators T(±) <sup>1</sup> can be easily deduced from such invariance properties. It is also straightforward to capture the notion that the natural degeneracy of the spectrum of the operator L<sup>1</sup> is the independence of its eigenvalues *<sup>λ</sup><sup>n</sup>* from the parameter *<sup>m</sup>*.

More precisely, because we have the following actions:

$$
\mathbb{T}\_1^{(-)} \psi\_{n, -\ell, -\ell} = \mathbb{T}\_1^{(+)} \psi\_{n\ell\ell} = 0,
$$

we can iterate the action of the lowering operator T(−) <sup>1</sup> so as to obtain:

$$\mathbb{T}\_1^{(-)}\Psi\_{n\ell\ell} = \mathbb{C}\_{\ell-1}\Psi\_{n,\ell,\ell-1}, \qquad \mathbb{T}\_1^{(-)}\mathbb{T}\_1^{(-)}\Psi\_{n\ell\ell} = \mathbb{C}\_{\ell-2}\Psi\_{n,\ell,\ell-2},$$

$$\cdots \cdot, \qquad (\mathbb{T}\_1^{(-)})^{\ell-1}\Psi\_{n\ell\ell} = \mathbb{C}\_{-\ell}\Psi\_{n,\ell,-\ell\ell}$$

or vice versa, by iterating the action of the raising operator T(+) <sup>1</sup> , the sequence:

$$\mathbb{T}\_1^{(+)}\psi\_{n,\ell,-\ell} = \mathbb{C}\_{-\ell+1}\psi\_{n,\ell,-\ell+1\prime} \qquad \qquad \mathbb{T}\_1^{(+)}\mathbb{T}\_1^{(+)}\psi\_{n,\ell,-\ell} = \mathbb{C}\_{-\ell+2}\psi\_{n,\ell,-\ell+2\prime}$$

$$\cdots \quad \quad \quad (\mathbb{T}\_1^{(+)})^{\ell-1}\psi\_{n,\ell,-\ell} = \mathbb{C}\_{\ell}\psi\_{n,\ell,+\ell}$$

where the coefficients *Ci* are coefficients of normalization, that is the actions of the raising and lowering operators T(±) <sup>1</sup> on the eigenfunctions *ψn*,-,*<sup>m</sup>* leave the parameters *n* and - unchanged and modify the parameter *m*, only.
