Baker-Akhiezer function for the deformed root system $BC(l,1)$ and bispectrality

TL;DR

This paper constructs a Baker-Akhiezer eigenfunction for a specific difference operator related to the deformed root system BC(l,1), establishing bispectral duality between Macdonald-Ruijsenaars and Calogero-Moser-Sutherland systems.

Contribution

It introduces a Baker-Akhiezer function for the BC(l,1) system and proves bispectral duality, extending results to complex multiplicities.

Findings

Existence of Baker-Akhiezer eigenfunction for the operator

Bispectral duality between MR and CMS systems of type BC(l,1)

Analytic continuation to complex multiplicities

Abstract

We show that a Sergeev-Veselov difference operator of rational Macdonald-Ruijsenaars (MR) type for the deformed root system $BC(l,1)$ preserves a ring of quasi-invariants in the case of non-negative integer values of the multiplicity parameters. We prove that in this case the operator admits a (multidimensional) Baker-Akhiezer eigenfunction, which depends on spectral parameters and which is, moreover, as a function of the spectral variables an eigenfunction for the (trigonometric) generalised Calogero-Moser-Sutherland (CMS) Hamiltonian for $BC(l,1)$. By an analytic continuation argument, we generalise this eigenfunction also to the case of more general complex values of the multiplicities. This leads to a bispectral duality statement for the corresponding MR and CMS systems of type $BC(l,1)$.