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.

翻译：本文分析了一类时间离散格式，用于在任意Lipschitz域 $\Omega \subset \mathbb{R}^d$（$d \le 3$）上求解具有非光滑势的低正则非线性薛定谔方程。研究表明，这些格式及其最优局部误差结构允许在解和势的正则性假设低于经典方法（如分裂法或指数积分器方法）要求的情况下实现收敛。此外，我们证明了在周期边界条件下，对于任意分数阶正Sobolev空间 $H^{r}$（$r \ge 0$），格式具有一阶和二阶收敛性，这超越了更常见的$L^2$或$H^\sigma (\sigma>\frac{d}{2})$误差分析框架。数值实验验证了我们的结果。