I noticed something interesting studying this Sturm-Liouville Problem:

$$ \frac{d}{dx}\left(\sqrt{(1-x^{2})}\frac{df}{dx} \right)+\frac{\left(n \alpha x+\alpha^2 x^{2} + \lambda\right)f}{\sqrt{(1-x^{2})}}= 0,$$ where $\alpha \in \mathbb{R}$ with periodic boundary conditions on $[-1,1]$. (Lambda is the eigenvalue) From basic St.-Liouville theory we know that the spectrum will be discrete cause the Sturm-Liouville operator is self-adjoint and the problem is regular.

Furthermore, we will have that $-\infty <\lambda_0 < \lambda_1 \le \lambda_2 < \lambda_3 \le .. \ .$

Now I noticed by extensive numerical calculations that for odd $n$ we will have that $\infty < \lambda_0 < \lambda_1 < \lambda_2 <...< \lambda_n,$ but after $\lambda_n$ all eigenvalues occur pairwise $\lambda_{n+1}=\lambda_{n+2}$. Is there a way to explicitely show this?

Edit: Due to two good questions, I want to add some information to the question: The eigenfunctions seem to a double eigenvalue seem to be even/odd functions respectively. I assume $\alpha >0$, but maybe it is worth studying $\alpha=0$ first (is similar to Legendre's ODE).

Edit2: I think I could reduce my problem to a Linear Algebra problem. Due to the very different kind of question this observation includes I asked a new question about this: https://mathoverflow.net/questions/177814/explain-eigenvalue-structure-of-these-sequences-of-matrices .