The goal of this talk is to present an almost global result on the long-time behavior of small solutions to the cubic nonlinear Schrödinger equation in the presence of an external non-generic potential. Roughly speaking, non-generic potentials are potentials that exhibit a resonance at the bottom of the spectrum of the associated linear Schrödinger operator. These resonances make the analysis more involved, as we will see. To handle these issues, we use the distorted Fourier transform and introduce a modified eigenfunction basis tailored to the non-generic setting. We then discuss the nonlinear analysis through what we call the Nonlinear Spectral Distribution (NSD), a tempered distribution that encodes the nonlinear interactions in distorted Fourier space. Finally, we examine some nonlinear estimates specific to the non-generic setting.