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 gˆ∨, where gˆ 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 gg-type quantum integrable models. The ψ  -system for the solutions associated with the fundamental representations of gg leads to Bethe ansatz equations associated with the affine Lie algebra gˆ. We also study the A2r(2) affine Toda field equation in massless limit in detail and find its Bethe ansatz equations as well as T–Q relations.