The Laplace operator, measure concentration, Gauss functions, and quantum mechanics

TL;DR

This paper explores a novel representation of the Schrödinger equation solutions using higher-dimensional functions, revealing measure concentration effects that simplify the operators involved, with potential applications in quantum chemistry computations.

Contribution

It introduces a measure concentration-based approach to approximate the Laplace operator in high-dimensional quantum systems, simplifying the analysis of electron interactions.

Findings

Operators can be approximated by the Laplace operator in large systems

Measure concentration effects facilitate decoupling of electron-electron interactions

Potential applications in iterative quantum chemistry methods

Abstract

We represent in this note the solutions of the electronic Schr\"odinger equation as traces of higher-dimensional functions. This allows to decouple the electron-electron interaction potential but comes at the price of a degenerate elliptic operator replacing the Laplace operator on the higher-dimensional space. The surprising observation is that this operator can without much loss again be substituted by the Laplace operator, the more successful the larger the system under consideration is. This is due to a concentration of measure effect that has much to do with the random projection theorem known from probability theory. The text is in parts based on the publications [Numer. Math. 146, 219--238 (2020)] and [SIAM J. Matrix Anal. Appl., 43, 464--478 (2022)] of the author and adapts the findings there to the needs of quantum mechanics. Our observations could for example find use in iterative methods that map sums of products of orbitals and geminals onto functions of the same type.