Kernel functions and Bäcklund transformations for relativistic Calogero-Moser and Toda systems
type: Item
instance of: scholarly article
title: Kernel functions and Bäcklund transformations for relativistic Calogero-Moser and Toda systems
zbMATH Open document ID: 1295.37020
DOI: 10.1063/1.4764878
author: Martin Hallnäs
author: Simon Ruijsenaars
published in: Journal of Mathematical Physics
publication date: 2014-01-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1206.3786
full work available at URL: https://dspace.lboro.ac.uk/2134/11825
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 37K35
Mathematics Subject Classification ID: 70H40
zbMATH DE Number: 6245470
zbMATH Keywords: Kernel function
zbMATH Keywords: Bäcklund transformation
zbMATH Keywords: Calogero-Moser
zbMATH Keywords: Toda
zbMATH Keywords: Hamiltonian
MaRDI profile type: Publication
OpenAlex ID: W1966465128
arXiv ID: 1206.3786
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review text: The authors of this interesting paper investigate the kernel functions associated with the quantum relativistic Calogero-Moser system as well as periodic and nonperiodic versions of the Toda system and their dual. The classical relativistic Calogero-Moser and Toda systems can be considered as Hamiltonian systems. Thus the elliptic version of this systems, which describe \(N\) interacting particles on a line or ring, includes the Poisson commuting Hamiltonians of the form NEWLINE\[NEWLINE S_k(x,p)=\sum\limits_{I\subset\{1,\ldots ,N\}, \;|I|=k} V_I(x)\prod\limits_{l\in I}e^{\beta p_l}, \;\;k=1,\ldots ,N, NEWLINE\]NEWLINE where \(\beta =1/mc\), with \(c\) the speed of light, and \(m>0\) is the particle rest mass. The interaction between particles is represented by the function NEWLINE\[NEWLINE V_I(x)=\prod\limits_{m\in I, \;n\notin I} f(x_m-x_n), NEWLINE\]NEWLINE where \(f(z)= s^{-1}(z)\bigl[ s(z+\rho )s(z-\rho ) \bigr]^{1/2}\), \(s(z)\) is the Weierstrass sigma function, and \(\rho \) is a coupling constant. The quantized version of the Calogero-Moser system for the elliptic case is represented by the commuting analytic difference operators NEWLINE\[NEWLINE\begin{multlined} \hat{S}_{k}(x)=\sum\limits_{I\subset\{1,\ldots ,N\}, \;|I|=k} \;\prod\limits_{m\in I, \;n\notin I} f_{-}(x_m-x_n) \prod\limits_{l\in I}e^{-i\hbar \beta\partial_{x_{l}}}\\ \prod\limits_{m\in I; \;n\notin I} f_{+}(x_m-x_n), \;\;k=1,\ldots ,N, \end{multlined}NEWLINE\]NEWLINE where \(\hbar \) is the Planck constant, and \(f_{\pm }(z)=[s(z\pm\rho )/s(z)]^{1/2}\). The authors introduce here the elliptic kernel functions satisfying \(N\) kernel identities of the form \([\hat{S}_{k}(x)-\hat{S}_{k}(-y)]\Psi (x,y)=0\), \(k=1,\ldots ,N\). The kernel functions \(\Psi (x,y)\) for the relativistic Toda systems obtained for the periodic and nonperiodic case are constructed by the hyperbolic gamma function. Just as in the quantum elliptic and hyperbolic cases, the authors switch to the quantum Toda case using the …
reviewed by: Dimitar A. Kolev
description: scientific article; zbMATH DE number 6245470