Kernel functions and Bäcklund transformations for relativistic Calogero-Moser and Toda systems (Q2872394)

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 positive parameters \(a_{+}\) and \(a_{-}\). Thus the modular-invariant formulas involving the hyperbolic and elliptic gamma functions are more readily expressed.NEWLINENEWLINEAn interesting result here is the proof that nonperiodic Toda kernel functions can be obtained as a limit of the periodic ones.NEWLINENEWLINEAnother problem of interest discussed in the paper is the notion of ``dual relativistic Toda systems'' which follow from the action-angle map for the nonperiodic Toda systems. The quantum version of the Poisson commuting Hamiltonians \(\hat{H}_k\) and the corresponding kernel functions built by the hyperbolic gamma function are also investigated here.NEWLINENEWLINESome interesting results for the dual Toda system exist as well. The Bäcklund transformations for the classical relativistic Calogero-Moser and Toda systems are considered along with the canonical transformations \((x,p)\to (y,q)\) that preserve the Poisson commuting Hamiltonians.NEWLINENEWLINEThe very interesting conclusion is that the special kernel functions admit a limit that yields generating functions of Bäcklund transformations for the classical relativistic Calogero-Moser and Toda systems. The authors obtain the nonrelativistic results which are a continuation of previous results in the literature.NEWLINENEWLINEThe paper contains a lot of other consequences concerning the existence of kernel functions in classical and quantum systems.