Regularity of many-body Schr\"odinger evolution equation and its application to numerical analysis

TL;DR

This paper proves that the many-body Schrödinger evolution equation exhibits mixed regularity under the Pauli Principle and introduces a new numerical approximation method that enhances computational efficiency in quantum chemistry.

Contribution

It extends the known regularity results from stationary to evolution Schrödinger equations and develops a novel approximation method leveraging mixed derivatives.

Findings

Proved mixed regularity for many-body evolution Schrödinger equation.

Generalized Strichartz estimates for the evolution problem.

Designed a new numerical approximation improving quantum chemistry computations.

Abstract

A decade ago, the mixed regularity of stationary many-body Schr\"o\-dinger equation has been studied by Harry Yserentant through the Pauli Principle and the Hardy inequality (Uncertainty Principle). In this article, we prove that the many-body evolution Schr\"odinger equation has a similar mixed regularity if the initial data $u_0$ satisfies the Pauli Principle. By generalization of the Strichartz estimates, our method also applies to the numerical approximation of this problem: based on these mixed derivatives, we design a new approximation which can hugely improve the computing capability especially in quantum chemistry.