Is the there a unique correspondence between the potential and bound state wave functions?

Question

I'm asking specifically for the schrodinger equation. Is there a unique correspondence between the energy eigenfunctions $\phi_i(x)$ and the potential term $U(x) = V(x)\phi(x)$? Furthermore, is this still true for a more general potential term that is a functional of $\phi(x)$ such as: $U(x) \equiv F(\phi(x))$? If not, I'm interested in what statements can be made.