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/0003002 · Journal of Physics A Mathematical and General

On BaxterQ-operators for the Toda chain

G. P. Pronko

Published Nov 9, 2000• 13 citations
arXivHTMLPDF

Abstract

We suggest a procedure for the construction of Baxter Q -operators for the Toda chain. Apart from the one-parametric family of Q -operators, considered in our recent paper (Pronko 2000 Commun. Math. Phys. to appear) we also give the method of construction of two basic Q -operators and the derivation of the functional relations for these operators. Also we have found the relation of the basic Q -operators with Bloch solutions of the quantum linear problem.

DOI
10.1088/0305-4470/33/46/308

Related papers

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

Baxter Q-operators for integrable DST chain

A. E. Kovalsky, G. P. Pronko

Following the procedure, described in the paper nlin.SI/0003002, for the integrable DST chain we construct Baxter Q-operators as the traces of monodromy of some M-operators, that act in quantum and auxiliary spaces. Within this procedure we obtain two basic M-operators and derive some functional relations between them such as intertwining relations and wronskian-type relations between two basic Q-operators.

arXiv:nlin/0203030•Mar 14, 2002•6 citations

Solution of Baxter equation for the $q$-Toda and Toda$_2$ chains by NLIE

O. Babelon, Karol Kajetan Kozlowski, Vincent Pasquier

We construct a basis of solutions of the scalar t-Q equation describing the spectrum of the q-Toda and Toda _2 2 chains by using auxiliary non-linear integral equations. Our construction allows us to provide quantisation conditions for the spectra of these models in the form of thermodynamic Bethe Ansatz-like equations.

arXiv:1804.01749•SciPost Physics•Oct 18, 2018•5 citations

Baxter operator and Baxter equation for $q$-Toda and Toda$_2$ chains

O. Babelon, Karol Kajetan Kozlowski, Vincent Pasquier

•May 22, 2018•5 citations

Relativistic Toda chain

G. P. Pronko, С. М. Сергеев

Investigated is the relativistic periodic Toda chain, to each site of which the ultra-local Weyl algebra is associated. Weyl's $q$ we are considering here is restricted to be inside the unit circle. Quantum Lax operators of the model are intertwined by six vertex $R$-matrix. Both independent Baxter's $Q$-operators are constructed explicitly as seria over local Weyl generators. The operator-valued wronskian of $Q$-s is also calculated.

arXiv:nlin/0009027•Sep 14, 2000•7 citations

Baecklund transformations and Baxter's Q-operator

Evgeny Konstantinovich Sklyanin

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.

arXiv:nlin/0009009•Sep 4, 2000•10 citations
We suggest a procedure for the construction of Baxter Q -operators for the Toda chain. Apart from the one-parametric family of Q -operators, considered in our recent paper (Pronko 2000 Commun. Math. Phys. to appear) we also give the method of construction of two basic Q -operators and the derivation of the functional relations for these operators. Also we have found the relation of the basic Q -operators with Bloch solutions of the quantum linear problem.
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
Solution of Baxter equation for the qqq-Toda and Toda2_22​ chains by NLIE
Baxter operator and Baxter equation for qqq-Toda and Toda2_22​ chains
Investigated is the relativistic periodic Toda chain, to each site of which the ultra-local Weyl algebra is associated. Weyl's qqq we are considering here is restricted to be inside the unit circle. Quantum Lax operators of the model are intertwined by six vertex RRR-matrix. Both independent Baxter's QQQ-operators are constructed explicitly as seria over local Weyl generators. The operator-valued wronskian of QQQ-s is also calculated.
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.