We develop the frozen Gaussian approximation (FGA) for the fractional Schr\"odinger equation in the semi-classical regime, where the solution is highly oscillatory when the scaled Planck constant $\varepsilon$ is small. This method approximates the solution to the Schr\"odinger equation by an integral representation based on asymptotic analysis and provides a highly efficient computational method for high-frequency wave function evolution. In particular, we revise the standard FGA formula to address the singularities arising in the higher-order derivatives of coefficients of the associated Hamiltonian flow that are second-order continuously differentiable or smooth in conventional FGA analysis. We then establish its convergence to the true solution. Additionally, we provide some numerical examples to verify the accuracy and convergence behavior of the frozen Gaussian approximation method.

翻译：本文针对半经典区域下的分数阶薛定谔方程发展了冻结高斯近似（FGA）方法，其中当普朗克常数$\varepsilon$较小时，解的振荡特性显著。该方法基于渐近分析，通过积分表示对薛定谔方程的解进行近似，为高频波函数演化提供了一种高效计算方法。我们特别修正了标准FGA公式以处理相关哈密顿流系数高阶导数中的奇异性——这些系数在经典FGA分析中要求二阶连续可微或光滑。随后我们证明了该方法收敛到真实解。此外，通过数值算例验证了冻结高斯近似方法的精度与收敛行为。