Suppose we have a fourth-order differential operator $L$ that we know is self-adjoint. Suppose also that the typical Sturmian boundary conditions are satisfied.
Is $L$ also Sturm Liouville? (This is true for second-order) That is, can it be put into Sturm Liouville form? If so, does this involve using integrating factors like the second-order analogue and/or is it tougher to do?