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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.00815 |
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| _version_ | 1866912112663592960 |
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| author | McWhinnie, Iain Rooke, Liam Vrabec, Martin |
| author_facet | McWhinnie, Iain Rooke, Liam Vrabec, Martin |
| contents | We show that a Sergeev-Veselov difference operator of rational Macdonald-Ruijsenaars (MR) type for the deformed root system $BC(l,1)$ preserves a ring of quasi-invariants in the case of non-negative integer values of the multiplicity parameters. We prove that in this case the operator admits a (multidimensional) Baker-Akhiezer eigenfunction, which depends on spectral parameters and which is, moreover, as a function of the spectral variables an eigenfunction for the (trigonometric) generalised Calogero-Moser-Sutherland (CMS) Hamiltonian for $BC(l,1)$. By an analytic continuation argument, we generalise this eigenfunction also to the case of more general complex values of the multiplicities. This leads to a bispectral duality statement for the corresponding MR and CMS systems of type $BC(l,1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00815 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Baker-Akhiezer function for the deformed root system $BC(l,1)$ and bispectrality McWhinnie, Iain Rooke, Liam Vrabec, Martin Mathematical Physics Exactly Solvable and Integrable Systems We show that a Sergeev-Veselov difference operator of rational Macdonald-Ruijsenaars (MR) type for the deformed root system $BC(l,1)$ preserves a ring of quasi-invariants in the case of non-negative integer values of the multiplicity parameters. We prove that in this case the operator admits a (multidimensional) Baker-Akhiezer eigenfunction, which depends on spectral parameters and which is, moreover, as a function of the spectral variables an eigenfunction for the (trigonometric) generalised Calogero-Moser-Sutherland (CMS) Hamiltonian for $BC(l,1)$. By an analytic continuation argument, we generalise this eigenfunction also to the case of more general complex values of the multiplicities. This leads to a bispectral duality statement for the corresponding MR and CMS systems of type $BC(l,1)$. |
| title | Baker-Akhiezer function for the deformed root system $BC(l,1)$ and bispectrality |
| topic | Mathematical Physics Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2406.00815 |