Completeness of answers of a differential equation

Question

I have read in my professor's lecture notes that the answers to homogenous wave equation (spatial part) are complete and we can expand any function in the interval of that equation based on them. I think this has to do something with Sturm-Liouville theorem but I don't know what is this exactly and where should I look. what does this theorem that an operators answers are complete comes from?

Answer 1

Yes, what you are saying is basically correct: well-posed Sturm-Liouville equations give you eigenfunction expansions that are complete. The expansions may be a discrete sum of eigenfunctions where the eigenvalue is a function of the index of the sum. Or the expansion may be an integral with respect to the eigenvalue parameter instead. Or the expansion may be a discrete sum plus an integral expansion.