Example of differential equation with periodic boundary conditions that has at least two simple eigenvalues

Question

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.

Answer 1

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. 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.

Answer 2

A double eigenvalue of problem (2) or (3) occurs precisely when there is a closed (spectral) gap. A generic potential $q$ has all its gaps open, so provides an example where all eigenvalues of both problems (2) and (3) are simple.

The Mathieu potential mentioned by Michael in his more concrete answer is a well known explicit potential that has all its gaps open.