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

## TL;DR

This paper establishes the asymptotic behavior of joint eigenfunctions in hyperbolic relativistic Calogero-Moser systems, confirming that particles exhibit soliton scattering for all particle numbers, extending previous results.

## Contribution

It determines the dominant asymptotics of eigenfunctions for arbitrary particle numbers, confirming soliton scattering behavior in the hyperbolic relativistic Calogero-Moser system.

## Key findings

- Confirmed soliton scattering for all N > 3
- Determined asymptotics of eigenfunctions in the hyperbolic case
- Extended previous results from N=2,3 to general N

## Abstract

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 $N>3$.