We explore the zero-temperature behavior of an assembly of bosons
interacting through a zero-range, attractive potential. Because the twobody
interaction admits a bound state, the many-body model is best
described by a Hamiltonian that includes the coupling between atomic
and molecular components. Due to the positive scattering length, the
low-density collection is expected to remain stable against collapse despite
the attraction between particles. Although a variational many-body analysis
indicates a collapsing solution with only a molecular component to
its condensate at low density, the expected atomic condensate solution
can be obtained if the chemical potential is allowed to be complex valued.
In addition to revealing two discrete eigenfrequencies associated with the
molecular case, an expansion in small oscillations quantifies the imaginary
part of the chemical potential as proportional to a coherent decay rate of
the atomic condensate into a continuum of collective phonon excitations
about the collapsing lower state.