  The highly excited super-Tonks-Girardeau (sTG) gas was recently observed to
be extremely stable in the presence of a weak dipolar repulsion. Here we reveal
the underlying reason for this mysterious phenomenon. By exactly solving the
trapped small clusters with both contact and dipolar interactions, we show that
the reason lies in the distinct spectral responses between sTG gas and its
decaying channel (bound state) when turn on a weak dipolar interaction.
Specifically, a tiny dipolar force can produce a visible energy shift for the
localized bound state, but can hardly affect the extended sTG branch. As a
result, the avoided level crossing between two branches is greatly modified in
both location and width in the parameter axis of coupling strength, leading to
a more (less) stable sTG gas for a repulsive (attractive) dipolar force. These
results, consistent with experimental observations, are found to robustly apply
to both bosonic and fermionic systems.
