Corrispondenza Classica/Quantistica in teorie di campo integrabili TTbar perturbate

The aim of this thesis is to study the ODE/IM correspondence, a stunning link between classical and quantum integrable models that can be formulated as an identification of the local conserved charges. In this work it will be shown that, given two models in ODE/IM correspondence, their TTbar perturbed versions are in correspondence as well. This will be done exploiting the fact that at the classical level the TTbar perturbation can be reinterpreted as a dynamical change of coordinates. Using this approach, it will be proved that the classical perturbed energy evolves with the perturbation parameter according to the same partial differential equation that is fulfilled by the quantum perturbed energy. Along the way, many important properties of Integrable Models and of the TTbar perturbation will be reviewed.