TBA for the Toda chain

TL;DR

This paper provides a direct derivation of Nekrasov-Shatashvili's quantization conditions for the Toda chain, linking them to solutions of nonlinear integral equations and the Baxter equation, with implications for integrable models.

Contribution

It offers a new derivation of quantization conditions for the Toda chain, connecting them to the Baxter equation and suggesting broader applicability to integrable models.

Findings

Quantization conditions expressed via nonlinear integral equations.

Connection established between Nekrasov-Shatashvili conditions and Baxter equation.

Framework applicable to a wide class of integrable models.

Abstract

We give a direct derivation of a proposal of Nekrasov-Shatashvili concerning the quantization conditions of the Toda chain. The quantization conditions are formulated in terms of solutions to a nonlinear integral equation similar to the ones coming from the thermodynamic Bethe ansatz. This is equivalent to extremizing a certain function called Yang's potential. It is shown that the Nekrasov-Shatashvili formulation of the quantization conditions follows from the solution theory of the Baxter equation, suggesting that this way of formulating the quantization conditions should indeed be applicable to large classes of quantized algebraically integrable models.