# Even and odd solutions for the Schrödinger equation

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?

• It seems like this should be derivable as a special case from the usual analysis of Hill's equation.
– Buzz
Jun 29, 2021 at 1:41

## 1 Answer

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, 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.

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

• Many thanks! That closes the case. In fact I am interested in the nonexistence of an even function. Is there a reference for this generic property? Jun 29, 2021 at 15:18
• For Mathieu's equation, the statement is referred to as Ince's Theorem - I've added a link to the corresponding section in the NIST Handbook. Not sure how this statement is referred to in the more general Hill's equation context. Jun 29, 2021 at 15:22
• Thanks, I'll check that. Jun 29, 2021 at 15:25