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?

  • $\begingroup$ For arbitrary metrics...? $\endgroup$ – ClassicStyle Feb 27 '16 at 6:52
  • $\begingroup$ 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 $\endgroup$ – Dac0 Feb 27 '16 at 10:04

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