Long time behavior of small solutions of NLS with non-generic potentials in one dimension

We consider the one-dimensional cubic nonlinear Schrödinger equation with a non-generic real-valued external potential $V$. We prove almost global-in-time quantitative bounds for small solutions. More precisely, small initial data of size $\varepsilon$ in a weighted Sobolev space give rise to solutions with the sharp decay rate $t^{-1/2}$ in $L^{\infty}_x$ up to time $\exp(\frac{1}{c\varepsilon^{2}})$. The main novelty of our result is that no additional symmetry assumption is imposed on $V$.
First, we use a modification of the standard distorted Fourier transform basis to resolve the possible discontinuity at zero energy due to the presence of a resonance. Then, following the work of Chen and Pusateri, we use smoothing estimates in the setting of non-generic potentials to analyze the low frequency structure of the (modified) nonlinear spectral distribution. A key novel ingredient is a Fourier restriction type inequality that handles low frequency contributions not amenable to the approach of Chen and Pusateri, and which is central to establishing the quantitative bounds.