Opers for Higher States of Quantum KdV Models

We study the ODE/IM correspondence for all states of the quantum g^ -KdV model, where g^ is the affinization of a simply-laced simple Lie algebra g. We construct quantum g^ -KdV opers as an explicit realization of the class of opers introduced by Feigin and Frenkel (Exploring new structures and natural constructions in mathematical physics, Math. Soc. Japan, Tokyo, 2011), which are defined by fixing the singularity structure at 0 and ∞, and by allowing a finite number of additional singular terms with trivial monodromy. We prove that the generalized monodromy data of the quantum g^ -KdV opers satisfy the Bethe Ansatz equations of the quantum g^ -KdV model. The trivial monodromy conditions are equivalent to a complete system of algebraic equations for the additional singularities.