  We study the Bethe Ansatz Equations for the Quantum KdV model, which are also
known to be solved by the spectral determinants of a specific family of
anharmonic oscillators called monster potentials (ODE/IM correspondence). These
Bethe Ansatz Equations depend on two parameters, identified with the momentum
and the degree at infinity of the anharmonic oscillators. We provide a complete
classification of the solutions with only real and positive roots -- when the
degree is greater than 2 -- in terms of admissible sequences of holes. In
particular, we prove that admissible sequences of holes are naturally
parameterised by integer partitions, and we prove that they are in one-to-one
correspondence with solutions of the Bethe Ansatz Equations if the momentum is
large enough. Consequently, we deduce that the monster potentials are complete,
in the sense that every solution of the Bethe Ansatz Equations coincides with
the spectrum of a unique monster potential. This essentially (i.e. up to gaps
in the previous literature) proves the ODE/IM correspondence for the Quantum
KdV model/monster potentials -- which was conjectured by Dorey-Tateo and
Bazhanov-Lukyanov-Zamolodchikov -- when the degree is greater than 2. Our
approach is based on the transformation of the Bethe Ansatz Equations into a
free-boundary nonlinear integral equation -- akin to the equations known in the
physics literature as DDV or KBP or NLIE -- of which we develop the
mathematical theory from the beginning.
