# Joint Eigenfunctions for the Relativistic Calogero–Moser Hamiltonians of Hyperbolic Type. III. Factorized Asymptotics Journal article, 2021

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 $N>3$.

## Author

### Martin Hallnäs

Chalmers, Mathematical Sciences, Analysis and Probability Theory

University of Gothenburg

### Simon Ruijsenaars

University of Leeds

#### International Mathematics Research Notices

1073-7928 (ISSN) 1687-0247 (eISSN)

Vol. 2021 6 4679-4708 rnaa193

### Quasi-invariants of finite Coxeter groups and integrable systems

Swedish Research Council (VR) (2018-04291), 2019-01-01 -- 2022-12-31.

Basic sciences

### Subject Categories

Other Mathematics

Mathematical Analysis

### DOI

10.1093/imrn/rnaa193