Airy functions in the thermodynamic Bethe ansatz

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.