Y-systems and wall crossing

The Thermodynamic Bethe Ansatz (TBA) equations have proved to be crucial to provide exact results in many areas of modern physics. The scope of its applications extends from lattice models for condensed matter physics to Integrable Quantum Field Theories (IQFTs). Research is also very active on its Ordinary Differential Equations (ODE) application, in particular, in the light of the correspondence between Integrable Models (IMs) and the ODEs associated to some classes of Schrödinger-like problems (ODE-IM correspondence). The possibility to formulate TBA systems in terms of functional relations known as Y-systems has brought to an alternative formulation of the problem and insights into the properties of its solutions. This thesis mainly focuses on the ODE side of the correspondence, and it investigates the formulation of Y-systems and TBA-systems of Schrödinger problems with polynomial potentials after wall crossing. Building on the results shown in Ito et al. (2019), an iterative procedure is proposed, so as to directly obtain the Y-system associated to a given chamber of the moduli space.