Airy Functions in the Thermodynamic Bethe Ansatz

Thermodynamic Bethe ansatz equations are coupled nonlinear 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 Painlevé III and the Toda hierarchy. In the simplest such case, the massless limit of these nonlinear 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 Schrödinger equation in a linear potential.