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

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.

arXiv:2411.15541 [quant-ph]

(or arXiv:2411.15541v3 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2411.15541 Focus to learn more arXiv-issued DOI via DataCite

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