We aim at analyzing a novel structure preserving difference scheme for the high dimensional nonlinear space-fractional Schrödinger equation. The temporal direction is discretized by the modified Crank-Nicolson scheme, while the spacial variable is approximated by formulas from the shifted convolution quadrature. The energy and mass preserving properties are proved rigorously for the scheme based on the newly developed properties of the coefficients matrix. Further, the optimal convergence rate $O(h^2+{\tau }^2)$ is derived in detail where $h,\tau$ denote the spacial and temporal mesh sizes, respectively. To save the computing cost, we consider the preconditioned fast BiCG (PF-BiCG) algorithm for the resulting complex systems, with the computing complexity of $O(N\log N)$ at each time step and the memory requirement of $O(N)$. Numerical experiments are conducted to confirm our theoretical conclusions and the efficiency of the fast algorithm.

我们旨在分析一种针对高维非线性空间分数分数薛定ding方程的新型结构保留差分格式。时间方向通过改进的Crank-Nicolson方案离散化，而空间变量则通过公式从位移卷积求积中近似得出。基于新开发的系数矩阵的性质，严格证明了该方案的能量和质量保持性质。此外，最佳收敛速度$Ø（H^2+{\tau }^2）$ 详细推导到 $H，\tau$分别表示空间和时间网格大小。为了节省计算成本，我们考虑将预处理的快速BiCG（PF-BiCG）算法用于生成的复杂系统，其计算复杂度为$Ø（ñ\mathrm{日志}ñ）$ 在每个时间步长和存储需求 $Ø（ñ）$。进行数值实验以证实我们的理论结论和快速算法的效率。