  We consider the $1d$ cubic nonlinear Schr\"odinger equation with a large
external potential $V$ with no bound states. We prove global regularity and
quantitative bounds for small solutions under mild assumptions on $V$. In
particular, we do not require any differentiability of $V$, and make spatial
decay assumptions that are weaker than those found in the literature (see for
example \cite{Del,N,GPR}). We treat both the case of generic and non-generic
potentials, with some additional symmetry assumptions in the latter case. Our
approach is based on the combination of three main ingredients: the Fourier
transform adapted to the Schr\"odinger operator, basic bounds on
pseudo-differential operators that exploit the structure of the Jost function,
and improved local decay and smoothing-type estimates. An interesting aspect of
the proof is an "approximate commutation" identity for a suitable notion of a
vectorfield, which allows us to simplify the previous approaches and extend the
known results to a larger class of potentials. Finally, under our weak
assumptions we can include the interesting physical case of a barrier potential
as well as recover the result of \cite{MMS} for a delta potential.
