Wall-crossing of TBA equations and WKB periods for the higher order ODE

We first study the WKB analysis of the third order ODE, which can be regarded as the quantized Seiberg-Witten curve of the (A2;AN)-type Argyres-Douglas theory in the Nekrasov-Shatashvili limit of Omega background. We then derive thermodynamic Bethe ansatz (TBA) equations satisfied by the Y-functions from the solutions of the ODE, and identify the Y-function with the WKB period. For the (A2;A2)-type ODE, we study the process of wall-crossing of the TBA equation from the minimal chamber to the maximal chamber.