On Generalized WKB Expansion of Monodromy Generating Function

TL;DR

This paper investigates the symplectic structure of monodromy maps for Schrödinger equations on Riemann surfaces, extending previous work by deriving a generalized WKB expansion of the Yang-Yang function and computing its initial terms.

Contribution

It introduces a generalized WKB expansion of the monodromy generating function, extending prior results to include higher-order terms in the expansion.

Findings

Derived conditions for the base projective connection to induce Goldman Poisson structure

Extended previous work by computing the first three terms of the WKB expansion of the Yang-Yang function

Established symplectic properties of the monodromy map in the context of meromorphic potentials

Abstract

We study symplectic properties of the monodromy map of the Schr\"odinger equation on a Riemann surface with a meromorphic potential having second-order poles. At first, we discuss the conditions for the base projective connection, which induces its own set of Darboux homological coordinates, to imply the Goldman Poisson structure on the character variety. Using this result, we extend the paper [Theoret. and Math. Phys. 206 (2021), 258-295, arXiv:1910.07140], by performing generalized WKB expansion of the generating function of monodromy symplectomorphism (the Yang-Yang function) and computing its first three terms.