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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2310.20481 |
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| _version_ | 1866917695648169984 |
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| author | Vieyra, J C Lopez Turbiner, A V |
| author_facet | Vieyra, J C Lopez Turbiner, A V |
| contents | One-dimensional 3-body Wolfes model with 2- and 3-body interactions also known as $G_2/I_6$-rational integrable model of the Hamiltonian reduction is exactly-solvable and superintegrable. Its Hamiltonian $H$ and two integrals ${\cal I}_{1}, {\cal I}_{2}$, which can be written as algebraic differential operators in two variables (with polynomial coefficients) of the 2nd and 6th orders, respectively, are represented as non-linear combinations of $g^{(2)}$ or $g^{(3)}$ (hidden) algebra generators in a minimal manner. By using a specially designed MAPLE-18 code to deal with algebraic operators it is found that $(H, {\cal I}_1, {\cal I}_2, {\cal I}_{12} \equiv [{\cal I}_1, {\cal I}_2])$ are the four generating elements of the {\it quartic} polynomial algebra of integrals. This algebra is embedded into the universal enveloping algebra $g^{(3)}$. In turn, 3-body/$A_2$-rational Calogero model is characterized by cubic polynomial algebra of integrals, it is mentioned briefly. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_20481 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Wolfes model aka $G_2/I_6$-rational integrable model: $g^{(2)}, g^{(3)}$ hidden algebras and quartic polynomial algebra of integrals Vieyra, J C Lopez Turbiner, A V Mathematical Physics Exactly Solvable and Integrable Systems Quantum Physics One-dimensional 3-body Wolfes model with 2- and 3-body interactions also known as $G_2/I_6$-rational integrable model of the Hamiltonian reduction is exactly-solvable and superintegrable. Its Hamiltonian $H$ and two integrals ${\cal I}_{1}, {\cal I}_{2}$, which can be written as algebraic differential operators in two variables (with polynomial coefficients) of the 2nd and 6th orders, respectively, are represented as non-linear combinations of $g^{(2)}$ or $g^{(3)}$ (hidden) algebra generators in a minimal manner. By using a specially designed MAPLE-18 code to deal with algebraic operators it is found that $(H, {\cal I}_1, {\cal I}_2, {\cal I}_{12} \equiv [{\cal I}_1, {\cal I}_2])$ are the four generating elements of the {\it quartic} polynomial algebra of integrals. This algebra is embedded into the universal enveloping algebra $g^{(3)}$. In turn, 3-body/$A_2$-rational Calogero model is characterized by cubic polynomial algebra of integrals, it is mentioned briefly. |
| title | Wolfes model aka $G_2/I_6$-rational integrable model: $g^{(2)}, g^{(3)}$ hidden algebras and quartic polynomial algebra of integrals |
| topic | Mathematical Physics Exactly Solvable and Integrable Systems Quantum Physics |
| url | https://arxiv.org/abs/2310.20481 |