TBA for the Toda chain

TL;DR

This work derives the Nekrasov–Shatashvili quantization conditions for the quantum Toda chain directly from the Baxter equation, by connecting the separation-of-variables framework to a nonlinear integral equation (NLIE) of the thermodynamic Bethe ansatz type. It develops two complementary routes: (i) Gutzwiller’s formulation via Hill determinants and pole data to obtain quantization conditions, and (ii) a NLIE-based approach that yields a TBA-like description in terms of a spectral parameter set $\delta$ and Yang’s potential $\mathcal W(\delta)$. The key contribution is showing that the NS quantization conditions emerge as extrema of $\mathcal W(\delta)$ and that the NLIE data can produce Baxter solutions $Q_{\delta}^{\pm}$, linking the NLIE, Wronskian relations, and the TQ equation. This framework suggests universality: similar NLIE–Baxter connections should apply to broad classes of quantized algebraically integrable models and provides a constructive path from spectral data to NS-type quantization.

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.

Theorems & Definitions (12)

• Proposition 1
• Lemma 1
• Lemma 2
• Proposition 2
• Lemma 3
• Proposition 3
• proof
• Lemma 4
• Proposition 4
• Proposition 5