On the 1d Cubic NLS with a Non-generic Potential

We consider the 1d cubic nonlinear Schrödinger equation with an external potential V that is non-generic. Without making any parity assumption on the data, but assuming that the zero energy resonance of the associated Schrödinger operator is either odd or even, we prove global-in-time quantitative bounds and asymptotics for small solutions. First, we use a simple modification of the basis for the distorted Fourier transform (dFT) to resolve the (possible) discontinuity at zero energy due to the presence of a resonance and the absence of symmetry of the solution. We then use a refined analysis of the low frequency structure of the (modified) nonlinear spectral distribution, and employ smoothing estimates in the setting of non-generic potentials.