Solving the linear semiclassical Schr\"odinger equation on the real line

TL;DR

This paper extends spectral splitting methods for solving the semiclassical Schr"odinger equation from the torus to the real line, enabling accurate numerical solutions for applications requiring unbounded domains.

Contribution

It introduces a symmetric Zassenhaus splitting method combined with spectral techniques tailored for the real line, advancing numerical solutions of the semiclassical Schr"odinger equation.

Findings

Developed a spectral splitting approach for  real line

Analyzed evolution of spectral bases under free Schr46dinger operator

Demonstrated improved accuracy in semiclassical regime

Abstract

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.