A general recursion for integrals involving products of Hermite polynomials and its applications
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866918351623684096 |
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| author | Anh-Tai, Tran Duong Son, Phan Quang Khang, Le Minh Vy, Nguyen Duy Pham, Vinh N. T. |
| author_facet | Anh-Tai, Tran Duong Son, Phan Quang Khang, Le Minh Vy, Nguyen Duy Pham, Vinh N. T. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_15541 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A general recursion for integrals involving products of Hermite polynomials and its applications Anh-Tai, Tran Duong Son, Phan Quang Khang, Le Minh Vy, Nguyen Duy Pham, Vinh N. T. Quantum Physics 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. |
| title | A general recursion for integrals involving products of Hermite polynomials and its applications |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2411.15541 |