  A correlated quantum many-body method is applied to describe resonance states
of atomic Bose-Einstein condensates (BEC) in a realistic shallow trap (as
opposed to infinite traps commonly used). The realistic van der Waals
interaction is adopted as the interatomic interaction. We calculate
experimentally measurable decay rates of the lowest quasi-bound state in the
shallow trap. The most striking result is the observation of a new metastable
branch besides the usual one for attractive BEC in a pure harmonic trap. As the
particle number increases the new metastable branch appears, then gradually
disappears and finally usual metastable branch (associated with the attractive
BEC in a harmonic trap) appears, eventually leading to the collapse of the
condensate.
