Three-body systems with Lagrange-mesh techniques in hyperspherical coordinates

We apply the Lagrange-mesh technique to three-body systems defined in hyperspherical coordinates. The method is applied to ${}^{6}\mathrm{He}$ (described as $\ensuremath{\alpha}+n+n)$ and ${}^{12}\mathrm{C}$ (described as $\ensuremath{\alpha}+\ensuremath{\alpha}+\ensuremath{\alpha}),$ and is shown to be fast and accurate. To deal with two-body forbidden states, we compare the usual projection method with the use of supersymmetric equivalent potentials. Both approaches provide similar spectroscopic properties, although the wave functions can be quite different. We also show that an accurate description of the three-body asymptotics requires bases with large hypermomenta.