Negative energy resonances of bosons in a magnetic quadrupole trap

We investigate resonances of spin-1 bosons in a three-dimensional magnetic quadrupole field. Complementary to the well-known positive energy resonances it is shown that there exist short-lived, i.e., broad, negative energy resonances. The latter are characterized by an atomic spin that is aligned antiparallel to the local magnetic field direction. In contrast to the positive energy resonances the lifetimes of the negative energy resonances decrease with increasing total magnetic quantum number. We derive a mapping of the two branches of the spectrum