Test of the slow variable discretization method in the adiabatic hyperspherical treatment of the $p+n+n$ system*

We consider the $p+n+n$ system using the Argonne $v_{18}$ plus the Urbana IX three-nucleon potential in the adiabatic hyperspherical description. Considering the $J=1/2+$ state, we solve for bound states and scattering properties in a two-step method. First, we use a hyperspherical harmonic expansion to calculate the adiabatic potential curves as a function of the hyperradius $R$. Second, we solve the remaining set of coupled equations in $R$ using Gauss-Lobatto basis function in a discrete variable representation together with a slow variable discretization of $R$ [O.~I. Tolstikhin, S. Watanabe, and M. Matsuzawa, J. Phys. B {\bf 29} L389 (1996)]. The resulting bound state energies not only agree well with benchmark calculations, but also show favorable convergence properties in comparison with the direct calculation of the coupling matrices. Two- and three-body scattering properties of the system are also calculated and extension to other scattering states and to the four-nucleon problem are discussed.