In this work we investigate the connection between discretized three-body continuum wave functions, in particular via a box boundary condition, and the wave functions computed with the correct asymptotics. The three-body wave functions are in both ca
ses obtained by means of the adiabatic expansion method. The information concerning all the possible incoming and outgoing channels, which appears naturally when the continuum is not discretized, seems to be lost when the discretization is implemented. In this work we show that both methods are fully equivalent, and the full information contained in the three-body wave function is actually preserved in the discrete spectrum. Therefore, in those cases when the asymptotic behaviour is not known analytically, i.e., when the Coulomb interaction is involved, the discretization technique can be safely used.
General expressions for the breakup cross sections in the lab frame for $1+2$ reactions are given in terms of the hyperspherical adiabatic basis. The three-body wave function is expanded in this basis and the corresponding hyperradial functions are o
btained by solving a set of second order differential equations. The ${cal S}$-matrix is computed by using two recently derived integral relations. Even though the method is shown to be well suited to describe $1+2$ processes, there are nevertheless particular configurations in the breakup channel (for example those in which two particles move away close to each other in a relative zero-energy state) that need a huge number of basis states. This pathology manifests itself in the extremely slow convergence of the breakup amplitude in terms of the hyperspherical harmonic basis used to construct the adiabatic channels. To overcome this difficulty the breakup amplitude is extracted from an integral relation as well. For the sake of illustration, we consider neutron-deuteron scattering. The results are compared to the available benchmark calculations.
