  We consider the asymptotics of the one-dimensional cubic nonlinear
Schr\"odinger equation with an external potential $V$ that does not admit bound
states. Assuming that $\langle x\rangle^{2+}V(x) \in L^1$ and that $u$ is
orthogonal to any zero-energy resonances, we show that solutions exhibit
modified scattering behavior. Our decay assumptions are weaker than those
appearing in previous work and are likely nearly optimal for exceptional
potentials, since the M\o{}ller wave operators are not known to be bounded on
$L^p$ spaces unless $\langle x\rangle^2V \in L^1$.
  Our method combines two techniques: First, we prove a new class of linear
estimates, which allow us to relate weights associated with the equation with
potential to weights associated with the flat equation, up to localized error
terms. By using the improved local decay coming from the zero-energy
assumption, we can treat these errors perturbatively, allowing us to treat the
weighted estimates using the same vector field arguments as in the flat case.
Second, we derive the asymptotic profile using the wave packet testing method
introduced by Ifrim and Tataru. Again, by taking advantage of improved local
decay near the origin, we are able to treat the potential as a lower-order
perturbation, allowing us to use tools developed to study the flat problem.
