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 \(J_N\) 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 \({}_2F_1\). 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 \(\psi _N\) of Baker–Akhiezer type, which are elementary, but highly nontrivial, functions. Specifically, we show that \(J_N\) essentially amounts to the antisymmetrisation of \(\psi _N\) and, as a byproduct, we obtain a recursive construction of \(\psi _N\) in terms of an iterated residue formula.