The numerical solution of a linear Schr\"odinger 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 Schr\"odinger 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 Schr\"odinger operator.
翻译:半古典体系中的线性 Schr\'odinger 方程式的数值解决方案在trus $\mathbb{T ⁇ d$中非常清楚。 现代计算方法的木筏是精确和负担得起的, 同时又能节省能量并解决高振荡。 但是,对于以$\mathb{R ⁇ d$( 一个更适合许多应用的设置 ) 的解决方案, 远非如此。 在本文中, 我们将分裂方法理论推广到此目的 。 主要的想法是从自由 Schr\“ odinger 方程式” 和直线弧普通差异方程式的解决方案组合中, 在一个对称的Zassenhaus 分裂法中, 从中得出光谱方法的解决方案。 这需要详细分析实际线上的某些异常光谱基础及其在自由 Schr\' odinger 操作器下的变化 。