2024/11/25 by Mironov, A., Morozov, A., Popolitov, A.
#FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.2411.16517
Within the context of wavefunctions of integrable many-body systems, rational multivariable Baker-Akhiezer (BA) functions were introduced by O. Chalykh, M. Feigin and A. Veselov and, in the case of the trigonometric Ruijsenaars-Schneider system, can be associated with a reduction of the Macdonald symmetric polynomials at t=q-m with integer partition labels substituted by arbitrary complex numbers. A parallel attempt to describe wavefunctions of the bispectral trigonometric Ruijsenaars-Schneider problem was made by M. Noumi and J. Shiraishi who proposed a power series that reduces to the Macdonald polynomials at particular values of parameters. It turns out that this power series also reduces to the BA functions at t=q-m, as we demonstrate in this letter. This makes the Macdonald polynomials, the BA functions and the Noumi-Shiraishi (NS) series a closely tied \it triad of objects, which have very different definitions, but are straightforwardly related with each other. In particular, theory of the BA functions provides a nice system of simple linear equations, while the NS functions provide a nice way to represent the multivariable BA function explicitly with arbitrary number of variables.