  A three-atom molecule AAB, formed by two identical bosons A and a distinct
one B, is studied by considering coupled channels close to a Feshbach
resonance. It is assumed that the subsystems AB and AA have, respectively, one
and two channels, where, in this case, AA has open and closed channels
separated by an energy gap. The induced three-body interaction appearing in the
single channel description is derived using the Feshbach projection operators
for the open and closed channels. An effective three-body interaction is
revealed in the limit where the trap setup is tuned to vanishing scattering
lengths . The corresponding homogeneous coupled Faddeev integral equations are
derived in the unitarity limit. The s-wave transition matrix for the AA
subsystem is obtained with a zero-range potential by a subtractive
renormalization scheme with the introduction of two finite parameters, besides
the energy gap. The effect of the coupling between the channels in the coupled
equations is identified with the energy gap, which essentially provides an
ultraviolet scale that competes with the van der Waals radius - this sets the
short-range physics of the system in the open channel. The competition
occurring at short distances exemplifies the violation of the ``van der Waals
universality" for narrow Feshbach resonances in cold atomic setups. In this
sense, the active role of the energy gap drives the short-range three-body
physics.
