# What are the eigenfunctions of the D'Alembert operator on pseudo-Riemannian manifolds?

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?

• For arbitrary metrics...? – ClassicStyle Feb 27 '16 at 6:52
• You can make the assumption you prefer on the metric, indeed I would like to have some estimate about the spectrum (if it's continuos, discrete, etc...). For example if something like Minakshisundaram result on Riemannian Manifolds hold also on Pseudoriemannian – Dac0 Feb 27 '16 at 10:04