Chalykh's Baker-Akhiezer functions as eigenfunctions of the integer-ray integrable systems

TL;DR

This paper introduces explicit Baker-Akhiezer functions as eigenfunctions of integer-ray integrable systems, revealing their simple difference equations, factorization properties, and generalizations to non-integer parameters and twisted cases.

Contribution

It provides explicit expressions for Baker-Akhiezer functions as eigenfunctions of integer-ray integrable systems, extending their applicability and understanding.

Findings

Explicit formulas for Baker-Akhiezer functions as eigenfunctions.

Demonstration of simple difference equations with constant coefficients.

Generalization to non-integer parameters and twisted cases.

Abstract

Macdonald symmetric polynomial at $t=q^{-m}$ reduces to a sum of much simpler complementary non-symmetric polynomials, which satisfy a simple system of the first order linear difference equations with constant coefficients, much simpler than those induced by the usual Ruijsenaars Hamiltonians of the cut-and-join type. We provide examples of explicit expressions for these polynomials nicknamed Baker-Akhiezer functions (BAF), and demonstrate that they further decompose into sums of nicely factorized quantities, perhaps, non-uniquely. Equations and solutions can be easily continued to non-integer parameters $\lambda$, which, in Macdonald polynomial case, are associated with integer partitions. Moreover, there is a straightforward generalization to "twisted" BAF's, which, however, are not so easy to decompose, and factorization of the coefficients is lost, at least naively. Still, these twisted BAF's provide eigenfunctions for Hamiltonians associated with commutative integer ray subalgebras of the Ding-Iohara-Miki algebra.