**3. Ladder Operators and the Degeneracy of the Spectrum of Operators**

We identify the degeneracy of the spectrum of the operator L<sup>1</sup> as a consequence of the existence of a particular kind of operators, called ladder operators. We provide a general definition of ladder operator after proving the following result, which can be indicated as the *shift theorem*.

**Theorem 5** (**Shift theorem**)**.** *Let* O *be an operator acting on a Hilbert space, and let* **v** *be an eigenfunction of* O *having an eigenvalue λ. If another operator* T *satisfies the condition* [O, T]**v** = *μ*T**v***, where the coefficient μ is a real number, then: either* T**v** *is the null function or* T**v** *is another eigenfunction of the operator* O *with eigenvalue λ* + *μ.*

**Proof.** If such an operator T exists, we have that, by linearity and and since *λ* is an eigenvalue of O, the above relation becomes:

$$[\mathbb{O}, \mathbb{T}]\mathbf{v} = \mu \mathbb{T}\mathbf{v} \qquad \Longleftrightarrow \qquad \mathbb{O}\mathbb{T}\mathbf{v} - \mathbb{T}\mathbb{O}\mathbf{v} = \mathbb{O}\mathbb{T}\mathbf{v} - \lambda\mathbb{T}\mathbf{v} = \mu \mathbb{T}\mathbf{v}$$

implying the new eigenvalue equation:

$$\mathbf{OTv} = (\lambda + \mu)\mathbf{Tv}\_{\prime}$$

meaning that either T**v** = **0** or *λ* + *μ* is an eigenvalue of O associated with the eigenfunction T**v**, so the proof is complete.

**Definition 4** (**Ladder operators**)**.** *An operator* T *satisfying the hypothesis of the shift theorem is called the ladder operator for the operator* O*. In particular,* T *is a:*


A very interesting case in which Theorem 5 is applied occurs when there exists a complete set of *n* self-adjoint operators O1, O2, ..., O*<sup>n</sup>* acting on a Hilbert space such that, by virtue of Theorem 3, there exists a basis of the space formed by all their simultaneous eigenfunctions {*y*1, *y*2,..., *yn*}.

If there exists an operator <sup>T</sup> commuting with the *<sup>k</sup>* operators <sup>O</sup>*i*<sup>1</sup> , <sup>O</sup>*i*<sup>2</sup> , ... , <sup>O</sup>*ik* and satisfying the *<sup>n</sup>* <sup>−</sup> *<sup>k</sup>* relations of the shift theorem with the remaining *<sup>n</sup>* <sup>−</sup> *<sup>k</sup>* operators <sup>O</sup>*j*<sup>1</sup> , <sup>O</sup>*j*<sup>2</sup> ,..., <sup>O</sup>*jn*−*<sup>k</sup>* for some certain eigenfunction *<sup>y</sup>*¯, the following relations hold:

$$\begin{cases} \mathbb{O}\_{i\_1}(\mathbb{T}\mathcal{Y}) = \mathbb{T}\mathbb{O}\_{i\_1}\mathcal{Y} = \mathbb{T}\lambda\_{i\_1}\mathcal{Y} = \lambda\_{i\_1}(\mathbb{T}\mathcal{Y})\\ \mathbb{O}\_{i\_2}(\mathbb{T}\mathcal{Y}) = \mathbb{T}\mathbb{O}\_{i\_2}\mathcal{Y} = \mathbb{T}\lambda\_{i\_2}\mathcal{Y} = \lambda\_{i\_2}(\mathbb{T}\mathcal{Y})\\ \vdots\\ \mathbb{O}\_{i\_k}(\mathbb{T}\mathcal{Y}) = \mathbb{T}\mathbb{O}\_{i\_k}\mathcal{Y} = \mathbb{T}\lambda\_{i\_k}\mathcal{Y} = \lambda\_{i\_k}(\mathbb{T}\mathcal{Y}) \end{cases} \tag{12a}$$

and:

$$[\mathbb{G}\_{\text{j}\prime}, \mathbb{T}]\bar{y} = \mu\_{\text{j}} \mathbb{T}\bar{y}, \quad [\mathbb{G}\_{\text{j}\prime}, \mathbb{T}]\bar{y} = \mu\_{\text{j}} \mathbb{T}\bar{y}, \quad \cdots, \quad [\mathbb{G}\_{\text{j}\prime}, \mathbb{T}]\bar{y} = \mu\_{\text{j}} \mathbb{T}\bar{y}, \tag{12b}$$

from which we obtain that the function T*y*¯ is either the null function or a simultaneous eigenfunction of <sup>O</sup>*i*<sup>1</sup> , <sup>O</sup>*i*<sup>2</sup> , ... , <sup>O</sup>*ik* with respect to the same eigenvalues *<sup>λ</sup>i*<sup>1</sup> , *<sup>λ</sup>i*<sup>2</sup> , ... , *<sup>λ</sup>ik* , respectively. Therefore, by Theorem 5, that function is a simultaneous eigenfunction of <sup>O</sup>*j*<sup>1</sup> ,..., <sup>O</sup>*jn*−*<sup>k</sup>* with respect to the shifted eigenvalues *<sup>λ</sup>j*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>j*<sup>1</sup> ,..., *<sup>λ</sup>jn*−*<sup>k</sup>* <sup>+</sup> *<sup>μ</sup>jn*−*<sup>k</sup>* .

**Remark 1.** *The degeneracy of the spectrum of a given operator* O *can be clarified (we precisely use the term 'clarification' if it is viewed in terms of ladder operators). Basically, we can consider the complete set of operators as a necessary tool to eliminate the degeneracy of the spectrum of* O *and to identify all the operators* T1, ... ,T*<sup>p</sup> that satisfy the relations* (12a) *together with the operator* O *and the relations* (12b) *of the shift theorem with the remaining operators of the complete set.*

The operator L<sup>1</sup> has a degenerate spectrum because its eigenvalues *<sup>λ</sup><sup>n</sup>* given in (11a) are independent of the parameters and *<sup>m</sup>*. Since the operator L<sup>1</sup> belongs to the complete set of operators {L1, *<sup>A</sup>*(*θ*, *<sup>ϕ</sup>*), <sup>M</sup><sup>1</sup> <sup>3</sup>}, in order to clarify the whole degeneracy in terms of ladder operators, it is sufficient to find the ladder operators <sup>T</sup><sup>1</sup> and <sup>T</sup><sup>2</sup> commuting with <sup>L</sup>1. Besides commuting with L1, such operators also satisfy the relations (12b) with the operators *<sup>A</sup>*(*θ*, *<sup>ϕ</sup>*) and <sup>M</sup><sup>1</sup> 3, in such a way that the functions <sup>T</sup>1*ψn<sup>m</sup>*(*r*, *<sup>θ</sup>*, *<sup>ϕ</sup>*) and T2*ψn<sup>m</sup>*(*r*, *θ*, *ϕ*) are eigenfunctions of <sup>L</sup><sup>1</sup> associated with the same eigenvalue *<sup>λ</sup><sup>n</sup>* and eigenfunctions of *<sup>A</sup>*(*θ*, *<sup>ϕ</sup>*) and of <sup>M</sup><sup>1</sup> <sup>3</sup> associated with a shifted eigenvalue with respect to and *m*, respectively.
