  In the description of the AdS5/CFT4 duality by an integrable system the
scattering matrix for bound states plays a crucial role: it was initially
constructed for the evaluation of finite size corrections to the planar
spectrum of energy levels/anomalous dimensions by the thermodynamic Bethe
ansatz, and more recently it re-appeared in the context of the glueing
prescription of the hexagon approach to higher-point functions. In this work we
present a simplified form of this scattering matrix and we make its pole
structure manifest. We find some new relations between its matrix elements and
also present an explicit form for its inverse. We finally discuss some of its
properties including crossing symmetry. Our results will hopefully be useful
for computing finite-size effects such as the ones given by the complicated
sum-integrals arising from the glueing of hexagons, as well as help towards
understanding universal features of the AdS5/CFT4 scattering matrix.
