I need an example of a periodic function $q:\mathbb{R} \to \mathbb{R}$ with period $\pi$ such that if we consider the differential equation 
\begin{equation}\tag{1}
y''(x)+(\lambda -q(x))y(x)=0
\end{equation} 
and the boundary conditions 
\begin{equation}\tag{2}
y(0)=y(\pi), \quad y'(0)=y'(\pi)
\end{equation}
\begin{equation}\tag{3}
y(0)=-y(\pi), \quad y'(0)=-y'(\pi),
\end{equation} 
then there exist at least two simple eigenvalues of the problem (1), (2) or there exist at least two simple eigenvalues of the problem (1), (3). 


I know that if $q(x)=0$ for all $x \in \mathbb{R}$, then (1), (2) has only a simple eigenvalue and all eigenvalues of (1), (3) are multiple.



Example of differential equation with periodic boundary conditions that has at least two simple eigenvalues
For the choice $q(x)=2q\cos (2x)$ (with $q$ on the right hand side a constant, I'm trying to stick to standard notation), the differential equation is known as Mathieu's equation, with solutions described, e.g., in the [NIST Handbook](https://dlmf.nist.gov/28.2#vi). The solutions for (1),(2) are the ones commonly denoted $ce\_n $, $se\_n $ for even $n$, the solutions for (1),(3) are the ones for odd $n$. The corresponding eigenvalues are simple as long as one keeps $q$ nonzero. For $q=0$, $ce\_n $, $se\_n $ indeed become degenerate, except for $n=0$, where one only has the solution $ce\_0 $, but no odd-parity counterpart.


