Teoria da ressurgência aplicada à mecânica quântica

Abstract

Divergent series arise in numerous cases from the perturbation theory of widespread use in physics. Those that are asymptotic can undergo treatments such as optimal truncation, which can be a good approximation, and Borel sum. However, a more difficult task is the reconstruction of functions from divergent perturbative expansions, as it generates incomplete information. This signals the need to include non-perturbative instantons effects. The reason for the misfortune of the divergence is related to the existence of singularities in the Borel complex plane, usually also associated with instantons. These are an important part in the construction of transeries, important manipulations of formal series that can cancel out ambiguities. An important effect of the resurgence theory is the relationship between the behavior of high orders of the perturbative expansion and the non-perturbative effects in low orders. From the mastery of methods to deal with asymptotic, from the understanding of non-perturbative effects of instantons and from the theory of resurgence for integrals with saddles, the application of resurgence in quantum mechanics is constructed for the Mathieu potential and his hyperbolic counterpart.