Expansion in terns of Laplacian eigenfunctions

Bobdemaths

Hi !

I am currently studying String theory in Polchinski's book. In section 6.2, eq. 6.2.2, he takes an arbitrary function $X(\sigma)$ defined on a Riemann Surface $M$. Then he expands it on a complete set of eigenfunctions of the laplacian,
$X(\sigma)=\sum_I x_I X_I(\sigma)$
with $\Delta X_I = - \omega_I^2 X_I$ and $\int_M d^2 \sigma \sqrt{g} X_I X_{I'} = \delta_{II'}$.

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 $I$ run in (discrete ? countable ?) ? It seems to be a very powerful tool, that's why I'm interested in those questions.

Thanks !

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