We consider $2a$ - periodic smooth solutions for
\begin{eqnarray\*}
-\Delta u+V(x)\,u=0\qquad\hbox{in}\:[-a,a]
\end{eqnarray\*}
We assume that $V$ is smooth and even (i.e. $V(-x)=V(x)$). We also assume that (up to multiplication with a real number) there exists only one odd $2a$ - periodic solution.
Can one say anything about the number of even $2a$ - periodic solutions?



Even and odd solutions for the Schrödinger equation
Because of the uniqueness of the initial value problem, there can be at most two solutions, i.e., if we have one odd $2a$-periodic solution, then there can be at most one more even $2a$-periodic solution. For example, for $V(x)=-(\pi /a)^2 $, we have the odd solution $\sin \pi x/a $ and the even solution $\cos \pi x/a $. On the other hand, for generic $V(x)$, the eigenvalues are simple, i.e., if we have an odd $2a$-periodic solution, there is no additional even $2a$-periodic solution. An example is the [Mathieu equation](https://dlmf.nist.gov/28.2#vi), rescaled to $2a$-periodicity, where $V(x)=(\pi^2/(2a)^2 ) (2q\cos (\pi x/a)-\lambda (q))$, with nonzero $q$ and associated eigenvalue $\lambda (q)$; cf.  [Ince's Theorem](https://dlmf.nist.gov/28.5#i).


In summary, there are either zero or one even $2a$-periodic solutions.


