My self-adjount differential operator $L$ is defined by $$L f(x) \equiv u(x) \frac{\partial^2}{\partial x^2} u(x) f(x)$$ where $u(x)$ is a known but arbitrary smooth function that satisfies $u(x) > 0$ $\forall x$.

I seek a basis of eigenfunctions of $L$, and their eigenvalues.

Are there any results known that help with this problem?

Are there any specific forms for $u(x)$ (other than $u\equiv 1$) that make the problem easier?