Transcendental Trace Formulas For Finite-Gap Potentials

TL;DR

This paper introduces new transcendental trace formulas involving modular functions and hypergeometric series for finite-gap potentials, expanding classical algebraic-geometric methods.

Contribution

It presents novel transcendental trace formulas that incorporate modular functions and hypergeometric series, differing from classical rational formulas in finite-gap integration.

Findings

Formulas involve transcendental modular functions

Derivation of transcendental relations for theta functions

Extension of classical trace formulas with hypergeometric series

Abstract

We show that formulas differing from classical analogues of rational trace formulas for algebraic-geometric potentials occur in the theory of finite-gap integration of spectral equations. The new formulas contain transcendental modular functions and hypergeometric series. They result in transcendental relations for theta functions.