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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2503.09502 |
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| _version_ | 1866917459599032320 |
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| author | Vieyra, Juan Carlos López Turbiner, Alexander V |
| author_facet | Vieyra, Juan Carlos López Turbiner, Alexander V |
| contents | An infinite 3-parametric family of superintegrable and exactly-solvable quantum models on a plane, admitting separation of variables in polar coordinates, marked by integer index $k$ was introduced in Journ Phys A 42 (2009) 242001 and was called in literature the TTW system. In this paper it is conjectured that the Hamiltonian and both integrals of TTW system have hidden algebra $g^{(k)}$ - it was checked for $k=1,2,3,4$ - having its finite-dimensional representation spaces as the invariant subspaces. It is checked that for $k=1,2,3,4$ that the Hamiltonian $H$, two integrals ${\cal I}_{1,2}$ and their commutator ${\cal I}_{12} = [{\cal I}_1,{\cal I}_2]$ are four generating elements of the polynomial algebra of integrals of the order $(k+1)$: $[{\cal I}_1,{\cal I}_{12}] = P_{k+1}(H, {\cal I}_{1,2},{\cal I}_{12})$, $[{\cal I}_2,{\cal I}_{12}] = Q_{k+1}(H, {\cal I}_{1,2},{\cal I}_{12})$, where $P_{k+1},Q_{k+1}$ are polynomials of degree $(k+1)$ written in terms of ordered monomials of $H, {\cal I}_{1,2},{\cal I}_{12}$. This implies that polynomial algebra of integrals is subalgebra of $g^{(k)}$. It is conjectured that all is true for any integer $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09502 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tremblay-Turbiner-Winternitz (TTW) system at integer index $k$: polynomial algebras of integrals Vieyra, Juan Carlos López Turbiner, Alexander V Mathematical Physics High Energy Physics - Theory Exactly Solvable and Integrable Systems An infinite 3-parametric family of superintegrable and exactly-solvable quantum models on a plane, admitting separation of variables in polar coordinates, marked by integer index $k$ was introduced in Journ Phys A 42 (2009) 242001 and was called in literature the TTW system. In this paper it is conjectured that the Hamiltonian and both integrals of TTW system have hidden algebra $g^{(k)}$ - it was checked for $k=1,2,3,4$ - having its finite-dimensional representation spaces as the invariant subspaces. It is checked that for $k=1,2,3,4$ that the Hamiltonian $H$, two integrals ${\cal I}_{1,2}$ and their commutator ${\cal I}_{12} = [{\cal I}_1,{\cal I}_2]$ are four generating elements of the polynomial algebra of integrals of the order $(k+1)$: $[{\cal I}_1,{\cal I}_{12}] = P_{k+1}(H, {\cal I}_{1,2},{\cal I}_{12})$, $[{\cal I}_2,{\cal I}_{12}] = Q_{k+1}(H, {\cal I}_{1,2},{\cal I}_{12})$, where $P_{k+1},Q_{k+1}$ are polynomials of degree $(k+1)$ written in terms of ordered monomials of $H, {\cal I}_{1,2},{\cal I}_{12}$. This implies that polynomial algebra of integrals is subalgebra of $g^{(k)}$. It is conjectured that all is true for any integer $k$. |
| title | Tremblay-Turbiner-Winternitz (TTW) system at integer index $k$: polynomial algebras of integrals |
| topic | Mathematical Physics High Energy Physics - Theory Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2503.09502 |