On Bäcklund transformations for many-body systems

Vadim Kuznetsov; Evgeni Sklyanin

doi:10.1088/0305-4470/31/9/012

On Bäcklund transformations for many-body systems

Vadim Kuznetsov, Evgeni Sklyanin

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

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 Bäcklund transformations (BTs) from the Hamiltonian point of view. The analogy between the BT and Baxter's quantum Q-operator pointed out by Pasquier and Gaudin is exploited to produce a conjugated variable for the parameter of the BT , such that belongs to the spectrum of the Lax operator . As a consequence, the generating function of the composition of n BTs gives rise to another canonical transformation separating variables for the model. For the Toda lattice the dual BT parametrized by is introduced.

Original language	English
Pages (from-to)	2241-2251
Number of pages	11
Journal	Journal of Physics A: Mathematical and General
Volume	31
Issue number	9
DOIs	https://doi.org/10.1088/0305-4470/31/9/012
Publication status	Published - Mar 1998

Keywords

Mathematical physics;
Statistical physics and nonlinear systems;

Access to Document

10.1088/0305-4470/31/9/012

http://arxiv.org/abs/solv-int/9711010

Cite this

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title = "On B{\"a}cklund transformations for many-body systems",

abstract = "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 B{\"a}cklund transformations (BTs) from the Hamiltonian point of view. The analogy between the BT and Baxter's quantum Q-operator pointed out by Pasquier and Gaudin is exploited to produce a conjugated variable for the parameter of the BT , such that belongs to the spectrum of the Lax operator . As a consequence, the generating function of the composition of n BTs gives rise to another canonical transformation separating variables for the model. For the Toda lattice the dual BT parametrized by is introduced. ",

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AU - Kuznetsov, Vadim

AU - Sklyanin, Evgeni

PY - 1998/3

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N2 - 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 Bäcklund transformations (BTs) from the Hamiltonian point of view. The analogy between the BT and Baxter's quantum Q-operator pointed out by Pasquier and Gaudin is exploited to produce a conjugated variable for the parameter of the BT , such that belongs to the spectrum of the Lax operator . As a consequence, the generating function of the composition of n BTs gives rise to another canonical transformation separating variables for the model. For the Toda lattice the dual BT parametrized by is introduced.

AB - 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 Bäcklund transformations (BTs) from the Hamiltonian point of view. The analogy between the BT and Baxter's quantum Q-operator pointed out by Pasquier and Gaudin is exploited to produce a conjugated variable for the parameter of the BT , such that belongs to the spectrum of the Lax operator . As a consequence, the generating function of the composition of n BTs gives rise to another canonical transformation separating variables for the model. For the Toda lattice the dual BT parametrized by is introduced.

KW - Mathematical physics;

KW - Statistical physics and nonlinear systems;

U2 - 10.1088/0305-4470/31/9/012

DO - 10.1088/0305-4470/31/9/012

M3 - Article

SN - 0305-4470

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SP - 2241

EP - 2251

JO - Journal of Physics A: Mathematical and General

JF - Journal of Physics A: Mathematical and General

IS - 9

ER -