We establish error bounds of the Lie-Trotter time-splitting sine pseudospectral method for the nonlinear Schr\"odinger equation (NLSE) with semi-smooth nonlinearity $ f(\rho) = \rho^\sigma$, where $\rho=|\psi|^2$ is the density with $\psi$ the wave function and $\sigma>0$ is the exponent of the semi-smooth nonlinearity. Under the assumption of $ H^2 $-solution of the NLSE, we prove error bounds at $ O(\tau^{\frac{1}{2}+\sigma} + h^{1+2\sigma}) $ and $ O(\tau + h^{2}) $ in $ L^2 $-norm for $0<\sigma\leq\frac{1}{2}$ and $\sigma\geq\frac{1}{2}$, respectively, and an error bound at $ O(\tau^\frac{1}{2} + h) $ in $ H^1 $-norm for $\sigma\geq \frac{1}{2}$, where $h$ and $\tau$ are the mesh size and time step size, respectively. In addition, when $\frac{1}{2}<\sigma<1$ and under the assumption of $ H^3 $-solution of the NLSE, we show an error bound at $ O(\tau^{\sigma} + h^{2\sigma}) $ in $ H^1 $-norm. Two key ingredients are adopted in our proof: one is to adopt an unconditional $ L^2 $-stability of the numerical flow in order to avoid an a priori estimate of the numerical solution for the case of $ 0 < \sigma \leq \frac{1}{2}$, and to establish an $ l^\infty $-conditional $ H^1 $-stability to obtain the $ l^\infty $-bound of the numerical solution by using the mathematical induction and the error estimates for the case of $ \sigma \ge \frac{1}{2}$; and the other one is to introduce a regularization technique to avoid the singularity of the semi-smooth nonlinearity in obtaining improved local truncation errors. Finally, numerical results are reported to demonstrate our error bounds.

翻译：针对具有半光滑非线性项$ f(\rho) = \rho^\sigma$的非线性薛定谔方程(NLSE)，其中$\rho=|\psi|^2$为波函数$\psi$的密度，$\sigma>0$为半光滑非线性指数，我们建立了Lie-Trotter时间分裂正弦伪谱方法的误差界。在NLSE具有$H^2$解的假设下，对于$0<\sigma\leq\frac{1}{2}$和$\sigma\geq\frac{1}{2}$两种情况，我们在$L^2$范数下分别证明了误差界为$ O(\tau^{\frac{1}{2}+\sigma} + h^{1+2\sigma}) $和$ O(\tau + h^{2}) $；对于$\sigma\geq \frac{1}{2}$的情况，在$H^1$范数下证明了误差界为$ O(\tau^\frac{1}{2} + h) $，其中$h$和$\tau$分别为网格尺寸和时间步长。此外，当$\frac{1}{2}<\sigma<1$且NLSE具有$H^3$解的假设下，我们在$H^1$范数下证明了误差界为$ O(\tau^{\sigma} + h^{2\sigma}) $。证明过程中采用两个关键策略：其一，针对$0<\sigma\leq\frac{1}{2}$情形，采用数值流的无条件$L^2$稳定性以避免对数值解的先验估计；针对$\sigma\ge\frac{1}{2}$情形，通过数学归纳法和误差估计建立$l^\infty$条件$H^1$稳定性以获得数值解的$l^\infty$界；其二，引入正则化技术以避免半光滑非线性在改进局部截断误差时产生的奇异性。最后，数值结果验证了所提出的误差界。