### Abstract

We investigate an N-state spin model called quantum relativistic Toda chain and based on the unitary finite-dimensional representations of the Weyl algebra with q being Nth primitive root of unity. Parameters of the finite-dimensional representation of the local Weyl algebra form the classical discrete integrable system. Nontrivial dynamics of the classical counterpart corresponds to isospectral transformations of the spin system. Similarity operators are constructed with the help of modified Baxter's Q-operators. The classical counterpart of the modified Q-operator for the initial homogeneous spin chain is a Bäcklund transformation. This transformation creates an extra Hirota-type soliton in a parameterization of the chain structure. Special choice of values of solitonic amplitudes yields a degeneration of spin eigenstates, leading to the quantum separation of variables, or the functional Bethe ansatz. A projector to the separated eigenstates is constructed explicitly as a product of modified Q-operators.

Original language | English |
---|---|

Pages (from-to) | 513-553 |

Number of pages | 41 |

Journal | International Journal of Mathematics and Mathematical Sciences |

Volume | 31 |

Issue number | 9 |

DOIs | |

Publication status | Published - 2002 |

Externally published | Yes |

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### Cite this

*International Journal of Mathematics and Mathematical Sciences*,

*31*(9), 513-553. https://doi.org/10.1155/S0161171202105059

}

*International Journal of Mathematics and Mathematical Sciences*, vol. 31, no. 9, pp. 513-553. https://doi.org/10.1155/S0161171202105059

**Quantum relativistic Toda chain at root of unity : Isospectrality, modified Q-operator, and functional bethe ansatz.** / Pakuliak, Stanislav; Sergeev, Sergei.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Quantum relativistic Toda chain at root of unity

T2 - Isospectrality, modified Q-operator, and functional bethe ansatz

AU - Pakuliak, Stanislav

AU - Sergeev, Sergei

PY - 2002

Y1 - 2002

N2 - We investigate an N-state spin model called quantum relativistic Toda chain and based on the unitary finite-dimensional representations of the Weyl algebra with q being Nth primitive root of unity. Parameters of the finite-dimensional representation of the local Weyl algebra form the classical discrete integrable system. Nontrivial dynamics of the classical counterpart corresponds to isospectral transformations of the spin system. Similarity operators are constructed with the help of modified Baxter's Q-operators. The classical counterpart of the modified Q-operator for the initial homogeneous spin chain is a Bäcklund transformation. This transformation creates an extra Hirota-type soliton in a parameterization of the chain structure. Special choice of values of solitonic amplitudes yields a degeneration of spin eigenstates, leading to the quantum separation of variables, or the functional Bethe ansatz. A projector to the separated eigenstates is constructed explicitly as a product of modified Q-operators.

AB - We investigate an N-state spin model called quantum relativistic Toda chain and based on the unitary finite-dimensional representations of the Weyl algebra with q being Nth primitive root of unity. Parameters of the finite-dimensional representation of the local Weyl algebra form the classical discrete integrable system. Nontrivial dynamics of the classical counterpart corresponds to isospectral transformations of the spin system. Similarity operators are constructed with the help of modified Baxter's Q-operators. The classical counterpart of the modified Q-operator for the initial homogeneous spin chain is a Bäcklund transformation. This transformation creates an extra Hirota-type soliton in a parameterization of the chain structure. Special choice of values of solitonic amplitudes yields a degeneration of spin eigenstates, leading to the quantum separation of variables, or the functional Bethe ansatz. A projector to the separated eigenstates is constructed explicitly as a product of modified Q-operators.

KW - quantum relativistic Toda chain

UR - http://www.scopus.com/inward/record.url?scp=0347792397&partnerID=8YFLogxK

U2 - 10.1155/S0161171202105059

DO - 10.1155/S0161171202105059

M3 - Article

VL - 31

SP - 513

EP - 553

JO - International Journal of Mathematics and Mathematical Sciences

JF - International Journal of Mathematics and Mathematical Sciences

SN - 0161-1712

IS - 9

ER -