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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.21775 |
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| _version_ | 1866911176785395712 |
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| author | Carpentier, Sylvain Mikhailov, Alexander V. Wang, Jing Ping |
| author_facet | Carpentier, Sylvain Mikhailov, Alexander V. Wang, Jing Ping |
| contents | We develop an algebraic quantisation approach, based on quantisation ideals, and apply it to integrable non-Abelian differential--difference equations. We show that the Toda hierarchy admits a bi-quantum structure whose classical (commutative) limit recovers a well-known Poisson pencil. In addition, we discover a non-standard quantisation that has no commutative counterpart. In both cases we present the quantum systems in the Heisenberg form.
The generality of the method is illustrated through a wide range of integrable lattices, including the modified Volterra, Bogoyavlensky, Ablowitz-Ladik, relativistic Toda, Merola-Ragnisco-Tu, Adler-Yamilov, Chen-Lee-Liu, Belov-Chaltikian, and Blaszak-Marciniak systems. For each of them, we construct explicit quantisation ideals and present the first few commuting quantum Hamiltonians. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_21775 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Algebraic quantisation approach to integrable differential-difference equations Carpentier, Sylvain Mikhailov, Alexander V. Wang, Jing Ping Exactly Solvable and Integrable Systems We develop an algebraic quantisation approach, based on quantisation ideals, and apply it to integrable non-Abelian differential--difference equations. We show that the Toda hierarchy admits a bi-quantum structure whose classical (commutative) limit recovers a well-known Poisson pencil. In addition, we discover a non-standard quantisation that has no commutative counterpart. In both cases we present the quantum systems in the Heisenberg form. The generality of the method is illustrated through a wide range of integrable lattices, including the modified Volterra, Bogoyavlensky, Ablowitz-Ladik, relativistic Toda, Merola-Ragnisco-Tu, Adler-Yamilov, Chen-Lee-Liu, Belov-Chaltikian, and Blaszak-Marciniak systems. For each of them, we construct explicit quantisation ideals and present the first few commuting quantum Hamiltonians. |
| title | Algebraic quantisation approach to integrable differential-difference equations |
| topic | Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2509.21775 |