Difference Operators and Duality for Trigonometric Gaudin and Dynamical Hamiltonians
We study the difference analog of the quotient differential operator from [Tarasov V., Uvarov F., Lett. Math. Phys. 110 (2020), 3375-3400, arXiv:1907.02117]. Starting with a space of quasi-exponentials =⟨αˣᵢᵢⱼ(), i = 1,…, , j = 1,…, ᵢ⟩, where αᵢ ∈ ℂ* and ᵢⱼ() are polynomials, we consider the formal...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2022 |
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| Sprache: | Englisch |
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Інститут математики НАН України
2022
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/211823 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Difference Operators and Duality for Trigonometric Gaudin and Dynamical Hamiltonians. Filipp Uvarov. SIGMA 18 (2022), 081, 41 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862723080436056064 |
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| author | Uvarov, Filipp |
| author_facet | Uvarov, Filipp |
| citation_txt | Difference Operators and Duality for Trigonometric Gaudin and Dynamical Hamiltonians. Filipp Uvarov. SIGMA 18 (2022), 081, 41 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We study the difference analog of the quotient differential operator from [Tarasov V., Uvarov F., Lett. Math. Phys. 110 (2020), 3375-3400, arXiv:1907.02117]. Starting with a space of quasi-exponentials =⟨αˣᵢᵢⱼ(), i = 1,…, , j = 1,…, ᵢ⟩, where αᵢ ∈ ℂ* and ᵢⱼ() are polynomials, we consider the formal conjugate Š† of the quotient difference operator ŠW satisfying Ŝ = ŠS. Here, S is a linear difference operator of order dim annihilating , and Ŝ is a linear difference operator with constant coefficients depending on αᵢ and degᵢⱼ() only. We construct a space of quasi-exponentials of dimension ord Š†, which is annihilated by Š† and describe its basis and discrete exponents. We also consider a similar construction for differential operators associated with spaces of quasi-polynomials, which are linear combinations of functions of the form ᶻ(), where ∈ ℂ and () is a polynomial. Combining our results with the results on the bispectral duality obtained in [Mukhin E., Tarasov V., Varchenko A., Adv. Math. 218 (2008), 216-265, arXiv:math.QA/0605172], we relate the construction of the quotient difference operator to the (ₖ, ₙ)-duality of the trigonometric Gaudin Hamiltonians and trigonometric dynamical Hamiltonians acting on the space of polynomials in anticommuting variables.
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| first_indexed | 2026-03-21T05:24:11Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211823 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T05:24:11Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
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| spelling | Uvarov, Filipp 2026-01-12T10:20:21Z 2022 Difference Operators and Duality for Trigonometric Gaudin and Dynamical Hamiltonians. Filipp Uvarov. SIGMA 18 (2022), 081, 41 pages 1815-0659 2020 Mathematics Subject Classification: 82B23; 17B80; 39A05; 34M35 arXiv:2202.06405 https://nasplib.isofts.kiev.ua/handle/123456789/211823 https://doi.org/10.3842/SIGMA.2022.081 We study the difference analog of the quotient differential operator from [Tarasov V., Uvarov F., Lett. Math. Phys. 110 (2020), 3375-3400, arXiv:1907.02117]. Starting with a space of quasi-exponentials =⟨αˣᵢᵢⱼ(), i = 1,…, , j = 1,…, ᵢ⟩, where αᵢ ∈ ℂ* and ᵢⱼ() are polynomials, we consider the formal conjugate Š† of the quotient difference operator ŠW satisfying Ŝ = ŠS. Here, S is a linear difference operator of order dim annihilating , and Ŝ is a linear difference operator with constant coefficients depending on αᵢ and degᵢⱼ() only. We construct a space of quasi-exponentials of dimension ord Š†, which is annihilated by Š† and describe its basis and discrete exponents. We also consider a similar construction for differential operators associated with spaces of quasi-polynomials, which are linear combinations of functions of the form ᶻ(), where ∈ ℂ and () is a polynomial. Combining our results with the results on the bispectral duality obtained in [Mukhin E., Tarasov V., Varchenko A., Adv. Math. 218 (2008), 216-265, arXiv:math.QA/0605172], we relate the construction of the quotient difference operator to the (ₖ, ₙ)-duality of the trigonometric Gaudin Hamiltonians and trigonometric dynamical Hamiltonians acting on the space of polynomials in anticommuting variables. The author would like to thank Vitaly Tarasov for many helpful discussions and for his valuable comments on a draft of this text. The author would like to thank the referees for their contribution to the improvement of the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Difference Operators and Duality for Trigonometric Gaudin and Dynamical Hamiltonians Article published earlier |
| spellingShingle | Difference Operators and Duality for Trigonometric Gaudin and Dynamical Hamiltonians Uvarov, Filipp |
| title | Difference Operators and Duality for Trigonometric Gaudin and Dynamical Hamiltonians |
| title_full | Difference Operators and Duality for Trigonometric Gaudin and Dynamical Hamiltonians |
| title_fullStr | Difference Operators and Duality for Trigonometric Gaudin and Dynamical Hamiltonians |
| title_full_unstemmed | Difference Operators and Duality for Trigonometric Gaudin and Dynamical Hamiltonians |
| title_short | Difference Operators and Duality for Trigonometric Gaudin and Dynamical Hamiltonians |
| title_sort | difference operators and duality for trigonometric gaudin and dynamical hamiltonians |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211823 |
| work_keys_str_mv | AT uvarovfilipp differenceoperatorsanddualityfortrigonometricgaudinanddynamicalhamiltonians |