Eigenvalues of an integral operator

[Peter Graf]

Hello,

I want to determine the eigenvalues of an integral
operator in particular of the operator
F(s)=int (d^2s' n_s \cdot (s-s')/abs(s-s')^3)
with s and s' are on the surface of an almost
spherical body and n_s is the normal vector in the
point s and the integration is performed over the
surface of the body.

(In order to clarify the operator I will just
verbalize the acting of the operator:
The argument of the operator is a three
dimensional vector, which lies on the surface of
an almost spherical body, the integration is
performed over the (2 dimensional) surface of the
body. The integrand is the scalar product of the
normal vector and the difference of s and s'
divided by the third power of the norm from (s-s'))

As the body is almost spherical my idea was to use
the spherical harmonics on order to calculate a
matrix of the operator and then determine the
eigenvalues of this matrix. However I wasn't able
to determine the matrix because of the singularity
of the integrand in the case s=s'.

So my question how could I numerical determine the
eigenvalues of F and/or how could I numerical
project the operator on the spherical harmonics.

Thanks in advance
Peter