A general recursion for integrals involving products of Hermite polynomials and its applications

TL;DR

This paper derives a simple, stable recursive formula for integrals of products of Hermite polynomials, enabling high-precision computations in physics and mathematics, especially for few-body quantum systems.

Contribution

It introduces a novel recursive scheme for Hermite polynomial integrals that avoids factorials, ensuring numerical stability and accuracy for large indices.

Findings

Provides a numerically stable recursive formula

Enables accurate computation of matrix elements in quantum simulations

Includes a practical subroutine implementation

Abstract

This study presents the derivation of a recursive formula for integrals of products of $N$ Hermite polynomials, establishing a numerically stable scheme for their accurate evaluation in computer codes. The derivation is notably simple and leverages solely the well-established properties of Hermite polynomials and the method of integration by parts. Importantly, our formulation completely circumvents explicit factorials, thereby preventing potential numerical instabilities and overflows, while facilitating high-precision computations for large indices. These findings are of significant relevance to a variety of areas in physics and mathematics. In particular, they offer an efficient and accurate framework for calculating two- and three-body matrix elements in ab initio simulations of few-body systems under a 1D harmonic confinement using the Configuration Interactions approach. A numerical subroutine implementing the recursive formula is provided as supplemental material.