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**Ackbeet** 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 $\displaystyle A^{2}$ is $\displaystyle A^{2}.$ That's what you'd be wanting to prove! I would try to show that the operator $\displaystyle D^{4}$, acting on any function satisfying your boundary conditions, is a self-adjoint operator. Then you know that the eigenvalues are real.