For pedagogical reasons, we shall start this review with very elementary reminders of the two-body problem, followed by some mathematical developments on the two-body Schrödinger equation, which will be useful for their three-body analogues discussed in Chapters 8, 9 and 10. The three-body problem is introduced in Chapter 3 through the simple but powerful harmonic-oscillator model. A natural continuation is the perturbed oscillator where anharmonicity corrections are treated to first order, leading to interesting regularities of the energy shifts: this will be presented in Chapter 8. In Chapter 4, we use the harmonic oscillator for the expansion of the wave function in the general case. This is the first of the variational methods we shall discuss.
Then we shall present in some detail the method of the hyperspherical expansion which, in many aspects, appears as a generalization of the harmonic oscillator, sharing most of its symmetries but relaxing the Gaussian character of the radial wave functions. Chapter 6 deals with the alternative method based on the Faddeev equations in configuration space, which can be applied very successfully to confining interactions.
Next comes a discussion on the quark–diquark approximation which is suited for angular excitations of ordinary baryons, and a discussion on the Born–Oppenheimer approximation which provides a simple picture of double-charm baryons QQq . While discussing several variational expansions or possible approximation schemes, we shall present numerical illustrations based on the simple potential model ∑ rijβ, with selected values of β and the same quark masses, so that the merits of the various methods can easily be compared.
In Chapter 8, we summarize the existing rigorous results on the level order of the three-body problem with confining interactions. We treat nearly harmonic potentials and nearly hyperscalar potentials and study how the zero-order degeneracy is broken. Then we compare the first radial and the first orbital excitations for a large class of local potentials.
Chapter 9 deals with mass inequalities. First, we analyse the mass dependence of the binding energy and derive inequalities between baryons of different flavour content which are bound by the same potential. Then we present inequalities which relate the binding energies of baryons and mesons when one assumes a simple relation between the quark antiquark potential in mesons and the potential between three quarks in baryons.
In Chapter 10, we present some bounds on the short-range correlations inside baryons. This is an important quantity for hyperfine splittings and weak decays. Unfortunately, the existing rigorous results are restricted to the case of equal-mass quarks. In Chapter 11, we apply the non-relativistic quark model to the phenomenology of the baryon spectrum and present various results on tests of flavour independence, the short-range character of the hyperfine interaction, etc.