Baker-Akhiezer specialisation of joint eigenfunctions for hyperbolic relativistic Calogero-Moser Hamiltonians

Martin HallnÃ¤s

paper

Baker-Akhiezer specialisation of joint eigenfunctions for hyperbolic relativistic Calogero-Moser Hamiltonians

arXiv:2509.23974

Abstract

In earlier joint work with Ruijsenaars, we constructed and studied symmetric joint eigenfunctions $J_{N}$ for quantum Hamiltonians of the hyperbolic relativistic $N$ -particle Calogero--Moser system. For generic coupling values, they are non-elementary functions that in the $N = 2$ case essentially amount to a `relativistic' generalisation of the conical function specialisation of the Gauss hypergeometric function $_{2} F_{1}$ . In this paper, we consider a discrete set of coupling values for which the solution to the joint eigenvalue problem is known to be given by functions $I ¨ _{N}$ of Baker--Akhiezer type, which are elementary, but highly nontrivial, functions. Specifically, we show that $J_{N}$ essentially amounts to the antisymmetrisation of $I ¨ _{N}$ and, as a byproduct, we obtain a recursive construction of $I ¨ _{N}$ in terms of an iterated residue formula.

19 pages. In v2 I have made some corrections, reorganized and expanded parts of the exposition and added a few references