Green's function of a differential operator contains a lot of information of that operator. In particular, if we have a differential operator on a compact manifold with discrete spectrum, then Green's function satisfies 


$$G(x,x',\lambda)=\sum\_n\frac{\psi\_n^\*(x)\psi\_n(x)}{\lambda-\lambda\_n}$$


where $\lambda\_n$ are the eigenvalues of the differential operator, $\psi\_n$ the corresponding eigenfunctions, and $\psi\_n^\*$ the complex conjugate.


In theory, we can look where the poles are in a Green's function and find the spectrum, and then calculate the residue of the poles to get the eigenfunctions. However, when an eigenvalue has multiplicity, then the residue might not give us the eigenfunctions in a straight-forward way. In this situation, is there any way to work around this difficulty and calculate the eigenfucntions?


More specifically, how can we write down a basis of the eigenspace associated to an eigenvalue, say 6, of the Laplacian on $S^3$ in terms of the Green's function of this operator?



Green's function and eigenvalues with multiplicity
I assume that your differential operator is linear unbounded with compact resolvent.
Eigenvalues of higher multiplicity have eigenspaces: any basis of the eigenspace form the eigenfunctions for this eigenvalue. They are not unique! But the expression in the Greens function is independent of the choice of an orthonormal basis of the eigenspace. 


