  We analyse a class of time discretizations for solving the nonlinear
Schr\"odinger equation with non-smooth potential and at low-regularity on an
arbitrary Lipschitz domain $\Omega \subset \mathbb{R}^d$, $d \le 3$. We show
that these schemes, together with their optimal local error structure, allow
for convergence under lower regularity assumptions on both the solution and the
potential than is required by classical methods, such as splitting or
exponential integrator methods. Moreover, we show first and second order
convergence in the case of periodic boundary conditions, in any fractional
positive Sobolev space $H^{r}$, $r \ge 0$, beyond the more typical $L^2$ or
$H^\sigma (\sigma>\frac{d}{2}$) -error analysis. Numerical experiments
illustrate our results.
