In this talk I will revise the basic ideas underlying a powerful correspondence between integrable models and ordinary differential equations (or IM/ODE correspondence, for short). This spectral equivalence played an important role in the study of PT-symmetry, providing rigorous proofs about the spectra of an important class of non-Hermitian Hamiltonians based on known integrable results. My intention is to make progress in the opposite direction, namely to extract information about integrable systems by using differential equations. The comparison between the eigenvalues of the ODEs and the spectral parameters in certain Quantum Integrable Models is considerably well understood for the vacuum state of the latter. For excited states, however, results are only known for the problem with su(2) symmetry. Here I will show this can be extended to the su(2|1) super-symmetric problem in order to establish the exact correspondence also for excited states.
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