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I am currently studying String theory in Polchinski's book. In section 6.2, eq. 6.2.2, he takes an arbitrary function [itex]X(\sigma)[/itex] defined on a Riemann Surface [itex]M[/itex]. Then he expands it on a complete set of eigenfunctions of the laplacian,

[itex]X(\sigma)=\sum_I x_I X_I(\sigma)[/itex]

with [itex]\Delta X_I = - \omega_I^2 X_I[/itex] and [itex]\int_M d^2 \sigma \sqrt{g} X_I X_{I'} = \delta_{II'}[/itex].

My question now is : when does such a decomposition exist ? Under which assumptions (on the initial function, and the Riemann surface) ? In what kind of set does [itex]I[/itex] run in (discrete ? countable ?) ? It seems to be a very powerful tool, that's why I'm interested in those questions.

Thanks !