We develop a refined Frozen Gaussian approximation (FGA) for the fractional Schr\"odinger equation in the semi-classical regime, where the solution exhibits rapid oscillations as the scaled Planck constant $\varepsilon$ becomes small. Our approach utilizes an integral representation based on asymptotic analysis, offering a highly efficient computational framework for high-frequency wave function evolution. Crucially, we introduce the momentum space representation of the FGA and a regularization parameter $\delta$ to address singularities in the higher-order derivatives of the Hamiltonian flow coefficients, which are typically assumed to be second-order differentiable or smooth in conventional analysis. We rigorously prove convergence of the method to the true solution and provide numerical experiments that demonstrate its precision and robust convergence behavior.

翻译：本文针对半经典区域内的分数阶薛定谔方程发展了一种改进的凝固高斯近似方法。当约化普朗克常数$\varepsilon$趋于零时，方程解呈现快速振荡特性。我们基于渐近分析构建了积分表示形式，为高频波函数演化提供了一个高效的计算框架。关键创新点在于引入了凝固高斯近似的动量空间表示以及正则化参数$\delta$，用以处理哈密顿流系数高阶导数中的奇异性问题——这类系数在传统分析中通常仅假设具有二阶可微性或光滑性。我们严格证明了该方法对真实解的收敛性，并通过数值实验验证了其精确度与鲁棒的收敛特性。