A fast solution method for time dependent multidimensional Schr\"odinger equations

TL;DR

This paper introduces a fast, high-order solution method for multidimensional Schrödinger equations using approximate basis functions, demonstrating efficiency in high-dimensional problems with separated representations.

Contribution

It presents a novel high-order approximation technique for multidimensional Schrödinger equations that is efficient in high dimensions and provides rigorous error estimates.

Findings

Effective in dimensions up to 200

Achieves approximation order up to 6

Negligible saturation error in computations

Abstract

In this paper we propose fast solution methods for the Cauchy problem for the multidimensional Schr\"odinger equation. Our approach is based on the approximation of the data by the basis functions introduced in the theory of approximate approximations. We obtain high-order approximations also in higher dimensions up to a small saturation error, which is negligible in computations, and we prove error estimates in mixed Lebesgue spaces for the inhomogeneous equation. The proposed method is very efficient in high dimensions if the densities allow separated representations. We illustrate the efficiency of the procedure on different examples, up to approximation order 6 and space dimension 200.