In this article we develop and analyse a new spectral method to solve the semiclassical Schrödinger equation based on the Gaussian wave-packet transform (GWPT) and Hagedorn’s semiclassical wave packets. The GWPT equivalently recasts the highly oscillatory wave equation as a much less oscillatory one (the $w$ equation) coupled with a set of ordinary differential equations governing the dynamics of the so-called GWPT parameters. The Hamiltonian of the $ w $ equation consists of a quadratic part and a small nonquadratic perturbation, which is of order $ \mathcal{O}(\sqrt {\varepsilon }) $, where $ \varepsilon \ll 1 $ is the rescaled Planck constant. By expanding the solution of the $ w $ equation as a superposition of Hagedorn’s wave packets, we construct a spectral method while the $ \mathcal{O}(\sqrt {\varepsilon }) $ perturbation part is treated by the Galerkin approximation. This numerical implementation of the GWPT avoids imposing artificial boundary conditions and facilitates rigorous numerical analysis. For arbitrary dimensional cases, we establish how the error of solving the semiclassical Schrödinger equation with the GWPT is determined by the errors of solving the $ w $ equation and the GWPT parameters. We prove that this scheme has spectral convergence with respect to the number of Hagedorn’s wave packets in one dimension. Extensive numerical tests are provided to demonstrate the properties of the proposed method.

中文翻译：

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基于高斯波包变换的半经典薛定谔方程的一种新的谱方法

在本文中，我们开发和分析了一种基于高斯波包变换 (GWPT) 和 Hagedorn 的半经典波包来求解半经典薛定谔方程的新谱方法。GWPT 等效地将高度振荡的波动方程改写为一个振荡少得多的波动方程（$w$ 方程），以及一组控制所谓 GWPT 参数动力学的常微分方程。$ w $ 方程的哈密顿量由一个二次部分和一个小的非二次扰动组成，其阶数为 $ \mathcal{O}(\sqrt {\varepsilon }) $，其中 $ \varepsilon \ll 1 $ 是重新缩放的普朗克常数。通过将 $ w $ 方程的解扩展为 Hagedorn 波包的叠加，我们构造了一个谱方法，而 $ \mathcal{O}(\sqrt {\varepsilon }) $ 扰动部分由 Galerkin 近似处理。GWPT 的这种数值实现避免了强加人为边界条件并促进了严格的数值分析。对于任意维度的情况，我们确定了用 GWPT 求解半经典薛定谔方程的误差如何由求解 $w$ 方程的误差和 GWPT 参数决定。我们证明了该方案对于一维的 Hagedorn 波包数量具有谱收敛性。提供了广泛的数值测试来证明所提出方法的特性。对于任意维度的情况，我们确定了用 GWPT 求解半经典薛定谔方程的误差如何由求解 $w$ 方程的误差和 GWPT 参数决定。我们证明了该方案对于一维的 Hagedorn 波包数量具有谱收敛性。提供了广泛的数值测试来证明所提出方法的特性。对于任意维度的情况，我们确定了用 GWPT 求解半经典薛定谔方程的误差如何由求解 $w$ 方程的误差和 GWPT 参数决定。我们证明了该方案对于一维的 Hagedorn 波包数量具有谱收敛性。提供了广泛的数值测试来证明所提出方法的特性。