Let $L$ be any differential operator (not necessarily linear).
Given initial conditions and boundary conditions (of any type), I am interested in general statements of the form:
Given a boundary value problem on some domain $\Omega$. If $L$, the initial condition, and the boundary are of a certain form, then separation of variables will yield the solution to the boundary value problem.
Do such general statements exist, and if they do, where can I find them? I am interested in statements about strong (classical) solutions and weak (distributional) solutions.
I am aware that questions similar to this one are already posted on this site, however, all of them consider specific differential operators and then they focus on separability of the domain. I am interested in statements about general differential operators.
Best Answer
The short answer is No. A major problem is that there is no single universal definition for what it means for an equation to be solvable by separation of variables. This defect in the theory, as well as the lack of general statements of the form that you would like to see, was highlighted a while ago in this BAMS book review:
Unfortunately, the situation has not changed so much since then. Koornwinder himself proposed a general definition for an equation to be separable and you can find a few papers that refer to it or extend it by looking up papers that cite this review. But there are no really general results, as far as I am aware.