线性半经典薛定谔方程在实直线上的求解
Solving the linear semiclassical Schrödinger equation on the real line
摘要：
半经典区域中线性薛定谔方程的数值解在环面$\mathbb{T}^d$中得到了很好的理解。大量的现代计算方法是精确和负担得起的，同时节省能源和解决高振荡非常好。然而，这与其在$\mathbb{R}^d$中的解决方案的情况相差甚远，$\mathbb{R}^d$是一种更适合于许多应用的设置。在本文中，我们扩展了分裂方法的理论。其主要思想是通过对称Zassenhaus分裂方法，从自由薛定谔方程和线性标量常微分方程的解的组合中，用谱方法得到解。这就需要详细分析实数线上的某些标准正交谱基及其在自由薛定谔算子下的演化。
英文摘要：
The numerical solution of a linear Schrödinger equation in the semiclassical regime is very well understood in a torus $\mathbb{T}^d$. A raft of modern computational methods are precise and affordable, while conserving energy and resolving high oscillations very well. This, however, is far from the case with regard to its solution in $\mathbb{R}^d$, a setting more suitable for many applications. In this paper we extend the theory of splitting methods to this end. The main idea is to derive the solution using a spectral method from a combination of solutions of the free Schrödinger equation and of linear scalar ordinary differential equations, in a symmetric Zassenhaus splitting method. This necessitates detailed analysis of certain orthonormal spectral bases on the real line and their evolution under the free Schrödinger operator.
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