  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 obtained 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.
