The building blocks of quantum observables are collections of asymptotic series in the coupling constant, corresponding to perturbative and non-perturbative sectors. According to the theory of resurgence, these series are related to each other in a nontrivial way, and these relations are encoded in the so-called Stokes automorphisms of the theory. In this seminar we put forward the idea that knowledge of these automorphisms and of the classical limit might make it possible to solve the quantum theory at all values of the coupling. We show how this program can be implemented in detail in quantum mechanics with polynomial potentials. In this case, the solution takes the form of a set of coupled integral equations of the TBA type, which generalizes the famous ODE/IM correspondence. This implementation uses ideas about BPS states and their wall-crossing in supersymmetric gauge theories.