Baker–Akhiezer Specialisation of Joint Eigenfunctions for Hyperbolic Relativistic Calogero–Moser Hamiltonians

Journal article, 2026

In earlier joint work with Ruijsenaars, we constructed and studied symmetric joint eigenfunctions JN 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 2F1. 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 ψN of Baker–Akhiezer type, which are elementary, but highly nontrivial, functions. Specifically, we show that JN essentially amounts to the antisymmetrisation of ψN and, as a byproduct, we obtain a recursive construction of ψN in terms of an iterated residue formula.