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 $ (or $\sigma = 1$). 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：本文针对具有低正则势及典型幂律非线性 $ f(\rho) = \rho^\sigma $（其中 $ \rho:=|\psi|^2 $ 为密度，$ \psi $ 为波函数，$ \sigma > 0 $ 为非线性指数）的非线性薛定谔方程，建立了时间分裂法的最优误差界。对于一阶Lie-Trotter时间分裂法，在 $L^\infty$-势与 $\sigma > 0$ 条件下证明了最优 $L^2$-范数误差界，并在 $W^{1,4}$-势与 $\sigma \geq 1/2$ 条件下得到最优 $H^1$-范数误差界。对于二阶Strang时间分裂法，在 $H^2$-势与 $\sigma \geq 1$ 条件下建立了最优 $L^2$-范数误差界，并在 $H^3$-势与 $\sigma \geq 3/2$（或 $\sigma = 1$）条件下证明了最优 $H^1$-范数误差界。与现有文献中时间分裂法的误差估计相比，我们的最优误差界或在相同正则性假设下提高了收敛阶数，或显著放宽了达到最优收敛阶所需的正则性要求。证明中的关键要素是采用称为"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，高频模式则通过解的正则性进行估计。大量数值结果验证了我们的误差估计，并表明其具有最优性。