General integral relations for the description of scattering states using the hyperspherical adiabatic basis

In this work we investigate 1+2 reactions within the framework of the hyperspherical adiabatic expansion method. To this aim two integral relations, derived from the Kohn variational principle, are used. A detailed derivation of these relations is shown. The expressions derived are general, not restricted to relative $s$ partial waves, and with applicability in multichannel reactions. The convergence of the $\mathcal{K}$-matrix in terms of the adiabatic potentials is investigated. Together with a simple model case used as a test for the method, we show results for the collision of a ${}^4$He atom on a \dimer dimer (only the elastic channel open), and for collisions involving a ${}^6$Li and two ${}^4$He atoms (two channels open).