Consider the operator $\Box=g^{\mu\nu}\nabla_\mu\nabla_\nu$ acting on a function space $\mathbf{F}(M)$, given by the set of functions $\phi:M\to\mathbb{R}$ whose values go to zero at infinity (at the boundary and outside of some compact region).
1 - What are the eigenfunctions of this operator?
2 - Is it possible to express any function as a linear combination of these eigenfunctions if the spectrum is continuous?