  Consider the nonlinear Schr\"odinger equation (NLS) with a potential with a
single negative eigenvalue. It has solitons with negative small energy, which
are asymptotically stable, and, if the nonlinearity is focusing, then also
solitons with positive large energy, which are unstable. In this paper we
classify the global dynamics below the second lowest energy of solitons under
small mass and radial symmetry constraints.
