Opers for higher states of quantum KdV models

We study the ODE/IM correspondence for all states of the quantum $\widehat{\mathfrak{g}}$-KdV model, where $\widehat{\mathfrak{g}}$ is the affinization of a simply-laced simple Lie algebra $\mathfrak{g}$. We construct quantum $\widehat{\mathfrak{g}}$-KdV opers as an explicit realization of the class of opers introduced by Feigin and Frenkel, which are defined by fixing the singularity structure at $0$ and $\infty$, and by allowing a finite number of additional singular terms with trivial monodromy. We prove that the generalized monodromy data of the quantum $\widehat{\mathfrak{g}}$-KdV opers satisfy the Bethe Ansatz equations of the quantum $\widehat{\mathfrak{g}}$-KdV model. The trivial monodromy conditions are equivalent to a complete system of algebraic equations for the additional singularities.