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 !

 

[AndyW]

Hello! It's great to see that you are studying String theory in Polchinski's book. The decomposition you are referring to is known as the spectral decomposition or spectral representation. It is a powerful tool in mathematics and physics, and it can be applied to various systems, including Riemann surfaces.

To answer your question, the decomposition exists under certain assumptions on the initial function and the Riemann surface. The function X(\sigma) must be square integrable on the Riemann surface, and the Riemann surface must have a finite area. Additionally, the eigenfunctions X_I(\sigma) must be orthogonal and complete, meaning that they form a basis for the space of square integrable functions on the Riemann surface.

The set I runs in is typically discrete, meaning that it consists of a finite or countably infinite number of indices. This is because the eigenvalues \omega_I^2 are discrete and the eigenfunctions form a discrete set. However, there are cases where the set I can be continuous, such as when the Riemann surface is a circle.

I hope this helps answer your questions. The spectral decomposition is indeed a powerful tool, and it has applications in various areas of mathematics and physics. Keep studying and exploring its uses in String theory. Best of luck!