Joint Eigenfunctions for the Relativistic Calogero–Moser Hamiltonians of Hyperbolic Type II. The Two- and Three-Variable Cases. (28th February 2017)

• Record Type: Journal Article
• Authors: Hallnäs, Martin; Ruijsenaars, Simon
• Is Part Of: International mathematics research notices. Volume 2018:Number 14(2018)
• Page Start: 4404
• Page End: 4449
• Publication Date: 2017-02-28
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnx020
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: 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 article, 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 ${\rm E}_2(b;x, y)$ and ${\rm E}_3(b;x, y)$ . In particular, we determine the dominant asymptotics for $y_1-y_2\to\infty$ and $y_1-y_2, y_2-y_3\to\infty$, respectively, from which the conjectured factorized scattering can be read off.