Your "Green's Formula" looks like the typical inner product used to show that a self-adjoint operator has real eigenvalues. You might want to double-check that formula, though, as you seem to be assuming that the adjoint of is That's what you'd be wanting to prove! I would try to show that the operator , acting on any function satisfying your boundary conditions, is a self-adjoint operator. Then you know that the eigenvalues are real.
Alternatively, you could go the long route: find the eigenvalues, and get a condition on them that must be true in order not to have trivial solutions (since eigenvectors aren't allowed, by definition, to be zero), and show that that condition implies the eigenvalues are real.
Those are my 3/4 baked thoughts.