Airy functions in the thermodynamic Bethe ansatz

TL;DR

This paper explores the connection between thermodynamic Bethe ansatz equations and differential equations, highlighting a unique closed-form solution involving Airy functions that relates to spectral determinants in quantum mechanics.

Contribution

It demonstrates that the massless limit of certain thermodynamic Bethe ansatz equations can be explicitly solved using Airy functions, linking integrable models to differential equations.

Findings

The massless limit solution is expressed in terms of Airy functions.

This solution is the only known closed-form for such TBA equations.

It relates the TBA equations to spectral determinants of Schrödinger operators.

Abstract

Thermodynamic Bethe ansatz equations are coupled non-linear integral equations which appear frequently when solving integrable models. Those associated with models with N=2 supersymmetry can be related to differential equations, among them Painleve III and the Toda hierarchy. In the simplest such case the massless limit of these non-linear integral equations can be solved in terms of the Airy function. This is the only known closed-form solution of thermodynamic Bethe ansatz equations, outside of free or classical models. This turns out to give the spectral determinant of the Schrodinger equation in a linear potential.