Integrable structures in resurgent quantum mechanics and wall-crossing

This Thesis is centered on the analysis of structures typical of integrable models that are found also in the context of the Schrödinger equation. Among the many relations that can be derived, particular focus is devoted to the study of the Thermodynamic Bethe Ansatz (TBA) equations, and the related Y-systems, for the Borel resummed version of the quantum periods associated to a given choice of potential function. These quantum periods are defined as loop integrals of formal solutions to the Riccati equation, which get promoted to actual functions through the resummation procedure. They are part of the monodromy data of the equation, and can be used to find exact quantisation conditions that allow to solve for the energy spectrum. Once the type of potential has been selected, the form of the TBA equations for the resummed quantum periods depends on the value chosen for the parameters entering in the potential; the parameter space may be in fact organised in regions, called chambers, differing for the type of TBA. Building on the results found by K. Ito et al. (2019), we have developed an algorithmic procedure that allows to find the TBA equations for the resummed quantum periods associated to a generic polynomial potential, and so in a generic chamber of the related parameter space, only through algebraic manipulations of the Y-system of a minimal chamber, where the form of the Y-system is always known. To further investigate the realm of applicability of this procedure, we then studied its possible implementation for the case of the modified Mathieu equation, a Schrödinger-like equation with a periodic potential, whose relevance lies in a connection with the deformation of the 4d N=2 supersymmetric SU(2) pure gauge theory.