# Rank-1 lattices and higher-order exponential splitting for the time-dependent Schr\"odinger equation

## TL;DR

This paper introduces a high-dimensional numerical method combining Fourier pseudo-spectral discretization on rank-1 lattices with higher-order exponential splitting for solving the time-dependent Schrödinger equation, achieving dimension-independent convergence.

## Contribution

The paper presents a novel approach that enables higher-order time convergence for high-dimensional Schrödinger equations using rank-1 lattice spatial discretization, independent of the problem dimension.

## Key findings

- Higher-order convergence demonstrated in 2 to 8 dimensions.
- Spatial discretization condition is independent of dimension.
- Numerical results confirm practical effectiveness.

## Abstract

In this paper, we propose a numerical method to approximate the solution of the time-dependent Schr\"odinger equation with periodic boundary condition in a high-dimensional setting. We discretize space by using the Fourier pseudo-spectral method on rank-$1$ lattice points, and then discretize time by using a higher-order exponential operator splitting method. In this scheme the convergence rate of the time discretization depends on properties of the spatial discretization. We prove that the proposed method, using rank-$1$ lattice points in space, allows to obtain higher-order time convergence, and, additionally, that the necessary condition on the space discretization can be independent of the problem dimension $d$. We illustrate our method by numerical results from 2 to 8 dimensions which show that such higher-order convergence can really be obtained in practice.