Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:nlin/0009009

Baecklund transformations and Baxter's Q-operator

Evgeny Konstantinovich Sklyanin

Published Sep 4, 2000• 10 citations
arXivHTMLPDF

Abstract

The course of 5 lectures given at the seminar "Integrable Systems: from Classical to Quantum" (Universite de Montreal, Jul 26 -- Aug 6, 1999) contains a detailed comment on the recently discovered (Gaudin-Pasquier, 1992) connection between Baecklund transformations in the theory of classical integrable systems on one hand, and Baxter's Q-operator for quantum integrable systems, on the other hand. We restrict our attention to the systems with finite number of degrees of freedom. Our main illustrative example is the periodic Toda lattice. We present a general construction of Q-operator for models governed by the SL(2)-invariant R-matrix and apply it to our example. We discuss also applications of BT and Q-operators to the separation of variables and theory of special functions.

DOI
10.48550/arxiv.nlin/0009009

Related papers

On Bäcklund transformations for many-body systems

V B Kuznetsov, E K Sklyanin

Using the n-particle periodic Toda lattice and the relativistic generalization due to Ruijsenaars of the elliptic Calogero-Moser system as examples, we revise the basic properties of the Baecklund transformations (BT's) from the Hamiltonian point of view. The analogy between BT and Baxter's quantum Q-operator pointed out by Pasquier and Gaudin is exploited to produce a conjugated variable mu for the parameter lambda of the BT B_lambda such that mu belongs to the spectrum of the Lax operator L(lambda). As a consequence, the generating function of the composition of n BT's gives rise also to another canonical transformation separating variables for the model. For the Toda lattice the dual BT parametrized by mu is introduced.

arXiv:solv-int/9711010•Journal of Physics A Mathematical and General•Mar 6, 1998•102 citations

Final report. Report for 1 Mar 78-30 Jun 82. [Baecklund transformations in classical and quantum mechanics]

David V. Chudnovsky, Gregory V. Chudnovsky

Using the method of Backlund transformations the investigators studied in detail classical and quantum two dimensional completely integrable systems, and lattice models equivalent to and approximating these continuous systems were constructed and examined. Among the continuous systems thoroughly examined in this way are the following important models of quantum field theory and statistical mechanics: the matrix (quantum and classical) nonlinear Schrodinger equation, the sine-Gordon equation, the matrix (non-Abelian) Toda lattice, the generalized XXX-and XXZ-quantum models and models of statistical mechanics, and the quantum generalized XYZ-lattice system (whose classical counterpart is the Landau-Lifshitz equation of superconductivity and whose statistical mechanics reductions include the eight-vertex model of Baxter.

•Jan 1, 1983•0 citations

Integrable systems : from classical to quantum : proceedings of the 38th session of the séminaire de mathématiques supérieures, July 26 - August 6, 1999 Montréal, Québec, Canada

J. Harnad, Gert Sabidussi, Pavel Winternitz A

On the chiral WZNW phase space, exchange r-matrices and Poisson-Lie groupoids by J. Balog, L. Feher, and L. Palla Loop groups, $R$-matrices and separation of variables by J. Harnad The geometry of generalised Hitchin systems by J. C. Hurtubise Determinant representation for form factors by V. E. Korepin Isomonodromic deformations in genus zero and one: Algebro-geometric solutions and Schlesinger transformations by D. A. Korotkin Quantum inverse scattering problem and correlation functions of integrable models by J.-M. Maillet Multiseparability and superintegrability for classical and quantum systems by W. Miller, Jr. Integrability and symmetry of the XXZ model by T. Miwa Characteristic systems on Poisson Lie groups and their quantization by N. Reshetikhin Special functions associated with Calogero-Moser type quantum systems by S. N. M. Ruijsenaars Backlund transformations and Baster's $Q$-operator by E. K. Sklyanin Universality of the distribution functions of random matrix theory by C. A. Tracy and H. Widom.

American Mathematical Society eBooks•Jan 1, 2000•0 citations

Baxter Operator and Baxter Equation for q-Toda and Toda2 Chains

O. Babelon, Karol Kajetan Kozlowski, Vincent Pasquier

We construct the Baxter operator [Formula: see text] for the [Formula: see text]-Toda chain and the Toda2 chain (the Toda chain in the second Hamiltonian structure). Our construction builds on the relation between the Baxter operator and Bäcklund transformations that were unravelled in [13]. We construct a number of quantum intertwiners ensuring the commutativity of [Formula: see text] with the transfer matrix of the models and of the [Formula: see text]’s between each other. Most importantly, [Formula: see text] is modular invariant in the sense of Faddeev. We derive the Baxter equation for the eigenvalues [Formula: see text] of [Formula: see text] and show that these are entire functions of [Formula: see text]. This last property will ultimately lead to the quantization of the spectrum for the considered Toda chains, in a subsequent publication [1]. This work is dedicated to the memory of L. D. Faddeev

arXiv:1803.06196•Reviews in Mathematical Physics•Jun 5, 2018•14 citations

DISCRETE KAUP–NEWELL EQUATION, CANONICAL BÄCKLUND TRANSFORMATION AND BAXTER'S Q-OPERATOR

A. Ghose Choudhury, Asesh Roy Chowdhury

We consider the Bäcklund transformation (BT) for a discretized version of the integrable Kaup–Newell equation, obtained recently by Tsuchida through a generalization of the Ablowitz–Ladik equation. The canonical nature of the classical Bäcklund transformation is proved by determining a suitable generating function. The corresponding Q-operator is determined by means of a quantization procedure proposed by Sklyanin et al. The latter's connection with Baxter's t - Q relation is explored and a comparison with the corresponding relation for the Bethe eigenvalue is made by solving the quantum inverse problem.

International Journal of Modern Physics A•Feb 10, 2002•2 citations

Baxter'sQ-operator for the homogeneousXXXspin chain

Sergey E. Derkachov

Applying the Pasquier-Gaudin procedure we construct Baxter's Q operator for the homogeneous XXX model as an integral operator in the standard representation of SL (2). The connection between the Q operator and the local Hamiltonians is discussed. We show that Lipatov's duality symmetry operator arises naturally as the leading term of the asymptotic expansion of the Q operator for large values of the spectral parameter.

arXiv:solv-int/9902015•Journal of Physics A Mathematical and General•Jan 1, 1999•96 citations
The course of 5 lectures given at the seminar "Integrable Systems: from Classical to Quantum" (Universite de Montreal, Jul 26 -- Aug 6, 1999) contains a detailed comment on the recently discovered (Gaudin-Pasquier, 1992) connection between Baecklund transformations in the theory of classical integrable systems on one hand, and Baxter's Q-operator for quantum integrable systems, on the other hand. We restrict our attention to the systems with finite number of degrees of freedom. Our main illustrative example is the periodic Toda lattice. We present a general construction of Q-operator for models governed by the SL(2)-invariant R-matrix and apply it to our example. We discuss also applications of BT and Q-operators to the separation of variables and theory of special functions.
Using the n-particle periodic Toda lattice and the relativistic generalization due to Ruijsenaars of the elliptic Calogero-Moser system as examples, we revise the basic properties of the Baecklund transformations (BT's) from the Hamiltonian point of view. The analogy between BT and Baxter's quantum Q-operator pointed out by Pasquier and Gaudin is exploited to produce a conjugated variable mu for the parameter lambda of the BT B_lambda such that mu belongs to the spectrum of the Lax operator L(lambda). As a consequence, the generating function of the composition of n BT's gives rise also to another canonical transformation separating variables for the model. For the Toda lattice the dual BT parametrized by mu is introduced.
Using the method of Backlund transformations the investigators studied in detail classical and quantum two dimensional completely integrable systems, and lattice models equivalent to and approximating these continuous systems were constructed and examined. Among the continuous systems thoroughly examined in this way are the following important models of quantum field theory and statistical mechanics: the matrix (quantum and classical) nonlinear Schrodinger equation, the sine-Gordon equation, the matrix (non-Abelian) Toda lattice, the generalized XXX-and XXZ-quantum models and models of statistical mechanics, and the quantum generalized XYZ-lattice system (whose classical counterpart is the Landau-Lifshitz equation of superconductivity and whose statistical mechanics reductions include the eight-vertex model of Baxter.
On the chiral WZNW phase space, exchange r-matrices and Poisson-Lie groupoids by J. Balog, L. Feher, and L. Palla Loop groups, RRR-matrices and separation of variables by J. Harnad The geometry of generalised Hitchin systems by J. C. Hurtubise Determinant representation for form factors by V. E. Korepin Isomonodromic deformations in genus zero and one: Algebro-geometric solutions and Schlesinger transformations by D. A. Korotkin Quantum inverse scattering problem and correlation functions of integrable models by J.-M. Maillet Multiseparability and superintegrability for classical and quantum systems by W. Miller, Jr. Integrability and symmetry of the XXZ model by T. Miwa Characteristic systems on Poisson Lie groups and their quantization by N. Reshetikhin Special functions associated with Calogero-Moser type quantum systems by S. N. M. Ruijsenaars Backlund transformations and Baster's QQQ-operator by E. K. Sklyanin Universality of the distribution functions of random matrix theory by C. A. Tracy and H. Widom.
We construct the Baxter operator [Formula: see text] for the [Formula: see text]-Toda chain and the Toda2 chain (the Toda chain in the second Hamiltonian structure). Our construction builds on the relation between the Baxter operator and Bäcklund transformations that were unravelled in [13]. We construct a number of quantum intertwiners ensuring the commutativity of [Formula: see text] with the transfer matrix of the models and of the [Formula: see text]’s between each other. Most importantly, [Formula: see text] is modular invariant in the sense of Faddeev. We derive the Baxter equation for the eigenvalues [Formula: see text] of [Formula: see text] and show that these are entire functions of [Formula: see text]. This last property will ultimately lead to the quantization of the spectrum for the considered Toda chains, in a subsequent publication [1]. This work is dedicated to the memory of L. D. Faddeev
We consider the Bäcklund transformation (BT) for a discretized version of the integrable Kaup–Newell equation, obtained recently by Tsuchida through a generalization of the Ablowitz–Ladik equation. The canonical nature of the classical Bäcklund transformation is proved by determining a suitable generating function. The corresponding Q-operator is determined by means of a quantization procedure proposed by Sklyanin et al. The latter's connection with Baxter's t - Q relation is explored and a comparison with the corresponding relation for the Bethe eigenvalue is made by solving the quantum inverse problem.