I will present recent progress in the amazing newly emerging subject of exact perturbation theory. This theory provides methods to upgrade divergent perturbative expansions to analytic information. One such method is called Borel resummation: it involves a detailed analysis (via a Borel transform) of a given divergent perturbative expansion in order to recover the nonperturbative corrections (i.e., exponentially small terms) that otherwise cannot be captured by the ordinary perturbation theory. The astounding big-picture upshot of this theory is that — contrary to Freeman Dyson’s conclusion that perturbation theory is incomplete — the divergent sector of perturbation theory actually already encodes all the necessary nonperturbative information, and it is therefore only a matter of applying suitable methods to extract it.
One of the most classical settings for exact perturbation theory is the so-called exact WKB method for solving singularly perturbed linear ODEs such as the Schrödinger equation. Such ODEs can be easily solved in (exponential) power series (the so-called WKB ansatz), but they are almost always divergent. Attempting to apply Borel resummation to get true analytic solutions turns out to be the correct approach, but a difficult mathematical problem. I will describe a solution I have developed through a series of recent works.