Joint eigenfunctions for the relativistic Calogero-Moser Hamiltonians of hyperbolic type. III. Factorized asymptotics

In the two preceding parts of this series of papers, we introduced and studied a recursion scheme for constructing joint eigenfunctions $J_N(a_+, a_-,b;x,y)$ of the Hamiltonians arising in the integrable $N$-particle systems of hyperbolic relativistic Calogero-Moser type. We focused on the first steps of the scheme in Part I, and on the cases $N=2$ and $N=3$ in Part II. In this paper, we determine the dominant asymptotics of a similarity transformed function \rE_N(b;x,y) for $y_j-y_{j+1}\to\infty$, $j=1,\ldots, N-1$, and thereby confirm the long standing conjecture that the particles in the hyperbolic relativistic Calogero-Moser system exhibit soliton scattering. This result generalizes a main result in Part II to all particle numbers $N>3$.Comment: 21 page

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