ODE/IM correspondence and Bethe ansatz for affine Toda field equations

We study the linear problem associated with modified affine Toda field equation for the Langlands dual ${\widehat{\mathfrak{g}}}^{\vee }$, where $\widehat{\mathfrak{g}}$ is an untwisted affine Lie algebra. The connection coefficients for the asymptotic solutions of the linear problem are found to correspond to the $Q$-functions for $\mathfrak{g}$-type quantum integrable models. The $\psi$-system for the solutions associated with the fundamental representations of $\mathfrak{g}$ leads to Bethe ansatz equations associated with the affine Lie algebra $\widehat{\mathfrak{g}}$. We also study the $A_{2r}^{(2)}$ affine Toda field equation in massless limit in detail and find its Bethe ansatz equations as well as T-Q relations.