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

We study the WKB periods for third order ordinary differential equation (ODE) with polynomial potential, which is obtained by Nekrasov-Shatashvili limit of ($A_2,A_N$) Argyres-Douglas theory in Omega background. In minimal chamber moduli space, we derive Y-system and thermodynamic Bethe ansatz (TBA) equations using ODE/IM correspondence. The exact are identified Y-functions. Varying parameters wall-crossing TBA occurs. process from to maximal $(A_2,A_2)$ $(A_2,A_3)$. When potential a monomial type, show ($A_2, A_2$) A_3$)-type ODE lead $D_4$ $E_6$-type respectively.