Shortly after the discovery of charmonium, in November 1974, several cc potentials were proposed, with an astonishing descriptive and predictive power. It was quickly realized that many different potentials V (r) can describe the main features of the data, namely the level ordering of the various radial and orbital excitations. This led Martin and others to search systematically sufficient conditions to be fulfilled by the potential in order to ensure a certain level ordering or a certain pattern of the level spacing. With the discovery of the upsilon family, the game (anticipated in fact by Eichten and Gottfried [74]) became even more challenging. One has to find classes of potentials describing different quarkonium families simultaneously, a tribute paid to the universality of the quark–antiquark potential. Many works have been published on the mass dependence of the binding energies and related tests of flavour independence.
The accumulation of new mathematical results on the confined two-body problem during the last 15 years is actually impressive. This encouraged some colleagues and myself to also investigate the three-body case. We checked our conjecture that the difficulty is far more severe than in the two-body case but we discovered, from the many papers on the quark model of baryons, that there are already a lot of partial results, just waiting to be collected together and generalized. This is why we are able to present here and in the next chapter a fair quantity of results on the three-body binding energies.