We establish optimal error bounds on time-splitting methods for the nonlinear Schr\"odinger equation with low regularity potential and typical power-type nonlinearity $ f(\rho) = \rho^\sigma $, where $ \rho:=|\psi|^2 $ is the density with $ \psi $ the wave function and $ \sigma > 0 $ the exponent of the nonlinearity. For the first-order Lie-Trotter time-splitting method, optimal $ L^2 $-norm error bound is proved for $L^\infty$-potential and $ \sigma > 0 $, and optimal $H^1$-norm error bound is obtained for $ W^{1, 4} $-potential and $ \sigma \geq 1/2 $. For the second-order Strang time-splitting method, optimal $ L^2 $-norm error bound is established for $H^2$-potential and $ \sigma \geq 1 $, and optimal $H^1$-norm error bound is proved for $H^3$-potential and $ \sigma \geq 3/2 $. Compared to those error estimates of time-splitting methods in the literature, our optimal error bounds either improve the convergence rates under the same regularity assumptions or significantly relax the regularity requirements on potential and nonlinearity for optimal convergence orders. A key ingredient in our proof is to adopt a new technique called \textit{regularity compensation oscillation} (RCO), where low frequency modes are analyzed by phase cancellation, and high frequency modes are estimated by regularity of the solution. Extensive numerical results are reported to confirm our error estimates and to demonstrate that they are sharp.

翻译：我们针对具有低正则势和典型幂律非线性项$ f(\rho) = \rho^\sigma $（其中$ \rho:=|\psi|^2 $为波函数$\psi$对应的密度，$\sigma > 0$为非线性指数）的非线性薛定谔方程，建立了时间分裂方法的最优误差界。对于一阶Lie-Trotter时间分裂方法，我们在$L^\infty$-势和$\sigma > 0$下证明了最优$L^2$-范数误差界，并在$W^{1,4}$-势和$\sigma \geq 1/2$下获得了最优$H^1$-范数误差界。对于二阶Strang时间分裂方法，我们在$H^2$-势和$\sigma \geq 1$下建立了最优$L^2$-范数误差界，并在$H^3$-势和$\sigma \geq 3/2$下证明了最优$H^1$-范数误差界。与文献中时间分裂方法的已有误差估计相比，我们的最优误差界要么在相同正则性假设下提高了收敛阶，要么显著放宽了实现最优收敛阶所需的势和非线性正则性要求。证明中的关键要素是采用一种称为\textit{正则补偿振荡}（RCO）的新技术，其中低频模式通过相位抵消分析，高频模式则利用解的正则性进行估计。我们报告了大量数值结果，以验证误差估计并证明其尖锐性。