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

Abstract In earlier joint work with Ruijsenaars, we constructed and studied symmetric joint eigenfunctions $$J_N$$ 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 $$N=2$$ N = 2 case essentially amount to a ‘relativistic’ generalisation of the conical function specialisation of the Gauss hypergeometric function $${}_2F_1$$ 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 $$\psi _N$$ ψ N of Baker–Akhiezer type, which are elementary, but highly nontrivial, functions. Specifically, we show that $$J_N$$ J N essentially amounts to the antisymmetrisation of $$\psi _N$$ ψ N and, as a byproduct, we obtain a recursive construction of $$\psi _N$$ ψ N in terms of an iterated residue formula.