Nested ansatz method for Baker-Akhiezer functions

TL;DR

This paper introduces a nested ansatz approach to derive Baker-Akhiezer functions, generalizing symmetrization techniques and resolving ambiguities in non-symmetric cases, including twisted scenarios, through recursive formulas and direct quantization.

Contribution

It presents a novel nested ansatz method for Baker-Akhiezer functions, extending existing techniques to non-symmetric and twisted cases with recursive formulas.

Findings

Successfully reproduces Noumi-Shiraishi formula for non-twisted BAFs

Resolves ambiguity issues in non-symmetric BAFs

Provides direct quantization results for twisted cases

Abstract

We explain that the logic behind the derivation of the Noumi-Shiraishi function can be applied directly to the Baker-Akhiezer function (BAF). This amounts to changing an ansatz for BAF to a nested one, where the BAF of N + 1 variables is recursively expressed as a sum over BAFs of N variables. This may be seen as a generalization of symmetrization trick from [1], but for the generally non-symmetric BAF. We demonstrate that, for usual non-twisted (a = 1) BAFs, this method correctly reproduces the Noumi-Shiraishi formula directly from linear equations, resolving the ambiguity related to non-simple roots. For the first non-trivial twisted case (N = 3, a = 2) this method also fixes this ambiguity, moreover, answers for the first few layers of coefficients are in the form of direct quantization of [1].