6.4 Results

We present below some results concerning the symmetric J = 0 case, i.e., the [56, 0⁺] states in our previous notation. We first notice that, for the harmonic oscillator, the partial-wave expansion is exactly restricted to angular momenta a = b = 0. It is thus useful to test the algorithms by solving Eq. (6.8) with V (ρ) = ρ².

Let us now consider the linear potential V (r) = 1 2 ∑ r_ij and analyse the results as a function of the maximal orbital momentum a_max introduced in the partial-wave expansion (6.7). A well-known trick consists of shifting the potential, to optimize the dominance of the a = 0 partial wave and ensure a nice asymptotic decrease of the Faddeev component Ψ₃ [66]. More precisely, one performs the computation with V = ∑ [1 2r_ij − v₀] and eventually adds 3v₀ to the eigenvalues. The results shown in Table 6.1 have been obtained with v₀ = 2. Preliminary investigations have confirmed that the results look much better for such large values of v₀.

The accuracy is not impressive for the eigenenergies, at least when they are computed as eigenvalues of the coupled integro-differential equations (6.8). This could probably be improved by recomputing the energy as  [66]. On the other hand, one should stress the impressive quality of the Faddeev wave function. With only the lowest partial wave a = b = 0, the correlation coefficient δ₁₂ is obtained almost exactly. We have seen in the previous chapters that it was very painful to achieve a decent convergence for δ₁₂ with variational methods.