Aspects of the inverse problem for the Toda chain

TL;DR

This paper extends methods for solving the inverse problem in the Toda chain by developing new operator equations in dual variables, leveraging monodromy matrix identities, and proposing a universal scheme applicable to various models solvable via quantum separation of variables.

Contribution

It generalizes Babelon's approach and applies Sklyanin's identities to derive new operator equations, broadening the applicability of inverse problem solutions in integrable models.

Findings

Developed new operator equations in dual variables for the Toda chain.

Applied monodromy matrix identities to derive equations for additional operators.

Proposed a universal scheme applicable to many models solvable by quantum separation of variables.

Abstract

We generalize Babelon's approach to equations in dual variables so as to be able to treat new types of operators which we build out of the sub-constituents of the model's monodromy matrix. Further, we also apply Sklyanin's recent monodromy matrix identities so as to obtain equations in dual variables for yet other operators. The schemes discussed in this paper appear to be universal and thus, in principle, applicable to many models solvable through the quantum separation of variables.