In this talk I will consider the ODE/IM correspondence for all states of the quantum gKdV model, where g is a simply laced affine Kac-Moody algebra. I will show how to construct quantum g-KdV opers as an explicit realization of a class of opers introduced by Feigin and Frenkel, which are defined by fixing the singularity structure at 0 and
infinity, and by allowing an arbitrary but finite number of additional singular terms with trivial monodromy. The generalized monodromy data of the quantum g-KdV opers satisfy the Bethe Ansatz equations of the quantum g-KdV model. In the sl2 case, the opers obtained are equivalent to the Schroedinger operators with “monster potential” obtained by Bazhanov, Lukyanov and Zamolodchikov in relation with the higher states of the quantum KdV model. Talk based on joint work with Davide Masoero.