Relativistic Calogero-Moser Research Articles

Joint Eigenfunctions for the Relativistic Calogero–Moser Hamiltonians of Hyperbolic Type II. The Two- and Three-Variable Cases

In a previous paper we introduced and developed a recursive construction of joint eigenfunctions $J_N(a_+,a_-,b;x,y)$ for the Hamiltonians of the hyperbolic relativistic Calogero-Moser system with arbitrary particle number $N$. In this paper we focus on the cases $N=2$ and $N=3$, and establish a number of conjectured features of the corresponding joint eigenfunctions. More specifically, choosing $a_+,a_-$ positive, we prove that $J_2(b;x,y)$ and $J_3(b;x,y)$ extend to globally meromorphic functions that satisfy various invariance properties as well as a duality relation. We also obtain detailed information on the asymptotic behavior of similarity transformed functions $\rE_2(b;x,y)$ and $\rE_3(b;x,y)$. In particular, we determine the dominant asymptotics for $y_1-y_2\to\infty$ and $y_1-y_2,y_2-y_3\to\infty$, resp., from which the conjectured factorized scattering can be read off.

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

We obtain kernel functions associated with the quantum relativistic Toda systems, both for the periodic version and for the nonperiodic version with its dual. This involves taking limits of previously known results concerning kernel functions for the elliptic and hyperbolic relativistic Calogero-Moser systems. We show that the special kernel functions at issue admit a limit that yields generating functions of Bäcklund transformations for the classical relativistic Calogero-Moser and Toda systems. We also obtain the nonrelativistic counterparts of our results, which tie in with previous results in the literature.

A Relativistic Conical Function and its Whittaker Limits

In previous work we introduced and studied a function $R(a_{+},a_{-},{\bf c};v,\hat{v})$ that is a generalization of the hypergeometric function ${}_2F_1$ and the Askey-Wilson polynomials. When the coupling vector ${\bf c}\in{\mathbb C}^4$ is specialized to $(b,0,0,0)$, $b\in{\mathbb C}$, we obtain a function ${\mathcal R}(a_{+},a_{-},b;v,2\hat{v})$ that generalizes the conical function specialization of ${}_2F_1$ and the $q$-Gegenbauer polynomials. The function ${\mathcal R}$ is the joint eigenfunction of four analytic difference operators associated with the relativistic Calogero-Moser system of $A_1$ type, whereas the function $R$ corresponds to $BC_1$, and is the joint eigenfunction of four hyperbolic Askey-Wilson type difference operators. We show that the ${\mathcal R}$-function admits five novel integral representations that involve only four hyperbolic gamma functions and plane waves. Taking their nonrelativistic limit, we arrive at four representations of the conical function. We also show that a limit procedure leads to two commuting relativistic Toda Hamiltonians and two commuting dual Toda Hamiltonians, and that a similarity transform of the function ${\mathcal R}$ converges to a joint eigenfunction of the latter four difference operators.

Zero-Eigenvalue Eigenfunctions for Differences of Elliptic Relativistic Calogero-Moser Hamiltonians

Letting A l (x) denote the commuting analytic difference operators of elliptic relativistic Calogero-Moser type, we present and study zero-eigenvalue eigenfunctions for the operators A l (x) - A l (-y) (with I = 1,2,..., N and x, y ∈ C N ). The eigenfunctions are products of elliptic gamma functions. They are invariant under permutations of x 1 , x N and y 1 ,..., y N and under interchange of the step-size parameters.

Isometric Reflectionless Eigenfunction Transforms for Higher-order AΔOs

In a previous paper (Regular and Chaotic Dynamics 7 (2002), 351–391, Ref. [1]), we obtained various results concerning reflectionless Hilbert space transforms arising from a general Cauchy system. Here we extend these results, proving in particular an isometry property conjectured in Ref. [1]. Crucial input for the proof comes from previous work on a special class of relativistic Calogero-Moser systems. Specifically, we exploit results on action-angle maps for the pertinent systems and their relation to the 2D Toda soliton tau-functions. The reflectionless transforms may be viewed as eigenfunction transforms for an algebra of higher-order analytic difference operators.

Reflectionless Analytic Difference Operators (AΔOs): Examples, Open Questions and Conjectures

We present a scenario concerning the existence of a large class of reflectionless self-adjoint analytic difference operators. In order to exemplify this scenario, we summarize our results on reflectionless self-adjoint difference operators of relativistic Calogero-Moser type.

Affine Toda solitons and systems of Calogero-Moser type

The solitons of affine Toda field theory are related to the spin-generalised Ruijsenaars-Schneider (or relativistic Calogero-Moser) models. This provides the sought after extension of the correspondence between the sine-Gordon solitons and the Ruijsenaars-Schneider model.

Relativistic Calogero-Moser model as gauged WZW theory

Integrable deformation of the Calogero-Moser system is examined in the framework of the topological G/ G Wess-Zumino-Witten model. It is shown that in the Hamiltonian approach the gauged WZW theory has a Hilbert space, which contains the one of the Ruijsenaars model. The latter can be described with the help of Verlinde algebra. Moreover, the evolution operator in the quantum mechanical problem has an interpretation in terms of the path integral in G/ G theory with inserted Wilson line. We compute a partition function of the model using techniques from Chem-Simons theory, in particular, some surgeries of simple threefolds.

Complete integrability of relativistic Calogero-Moser systems and elliptic function identities

Poincare-invariant generalizations of the Galilei-invariant Calogero-Moser AΓ-particle systems are studied. A quantization of the classical integrals S1?...,SN is presented such that the operators Si9...,SN mutually commute. As a corollary it follows that Si9...9SN Poisson commute. These results hinge on functional equations satisfied by the Weierstrass σ- and 0*- functions. A generalized Cauchy identity involving the σ-function leads to an N x N matrix L whose symmetric functions are proportional to Sl5 ...9SN.