Modelli Integrabili e problemi spettrali in Equazioni Differenziali Ordinarie

This thesis studies a particular (1+1)-dimensional integrable scattering theory, which generalizes the Lee-Yang theory with an additional Z_5 symmetry, then extended to Z_s where s is the spin of an arbitrary locally conserved charge in the Lee-Yang model. Thermodynamic Bethe ansatz method is performed and numerical results are compared with the perturbations near the corresponding conformal theory. The spectrum and pseudo-particle fusion rules are found. In the TBA content, functional equations, called Y-systems are defined. These allow the problem to be associated with a particular ODE/IM Correspondence case.