  We consider a multichannel generalization of the Fermi pseudopotential to
model low-energy atom-atom interactions near a magnetically tunable Feshbach
resonance, and calculate the adiabatic hyperspherical potential curves for a
system of three such interacting atoms. In particular, our model suggests the
existence of a series of quasi-bound Efimov states attached to excited
three-body thresholds, far above open channel collision energies. We discuss
the conditions under which such states may be supported, and identify which
interaction parameters limit the lifetime of these states. We speculate that it
may be possible to observe these states using spectroscopic methods, perhaps
allowing for the measurement of multiple Efimov resonances for the first time.
