%****************************************************************
\section{Summary}
%****************************************************************
In the first section of this chapter, we have reported exact solutions of a Coulomb correlation problem, consisting of two electrons on a $D$-dimensional sphere.
The Coulomb problem can be solved exactly for an infinite set of values of the radius $R$ for both the ground and excited states, on both the singlet and triplet manifolds.  
The corresponding exact solutions are polynomials in the interelectronic distance $\ree$.
The cusp conditions, which are related to the behaviour of the wave function at the electron-electron coalescence point, have been analysed and classified according to the natural or unnatural parity of the state considered.

In the second section, we proved that the leading term in the large-$D$ expansion of the high-density correlation energy of an electron pair is invariant to the nature of the confining potential.  
For any such system, the correlation energy is given by $\Ec \sim -\gamma^2/8$, where $\gamma = 1/(D-1)$ is the Kato cusp factor in a $D$-dimensional space.

