In my book I need to prove the general theorem that I can always pick the eigenstates in the time independent Schrödinger equation to be either even or odd ψ(x). This provides that the potential is an even function and is not hard to show if you use that if ψ(x) is a solution then too is ψ(-x) and you can write any solutions a superposition of an even and odd solutions. But my problem is that ψ are eigenstates. So if a general ψ which is neither even nor odd is found to be an eigenstate by solving the equation. How can you then say that the even and odd solution forming ψ are also eigenstates? For me that just doesn't make sense in terms of the discrete picture. Imagine you have an eigenvector a. You can scale it and pick a scaling of a as an eigenvector yes. But how would you possibly be able to write that eigenstate up as a linear combination of two other eigenstates? - the eigenstates are linearly independent for a hermitian operator!!! Thus I think I am misunderstanding something somewhere.