The singularly perturbed Riccati equation is the first-order nonlinear ordinary differential equation $\hbar \partial _x f = af^2 + bf + c$ in the complex domain where $\hbar $ is a small complex parameter. We prove an existence and uniqueness theorem for exact solutions with prescribed asymptotics as $\hbar \to 0$ in a half-plane. These exact solutions are constructed using the Borel–Laplace method; that is, they are Borel summations of the formal divergent $\hbar $ -power series solutions. As an application, we prove existence and uniqueness of exact WKB solutions for the complex one-dimensional Schrödinger equation with a rational potential.

We prove a functorial correspondence between a category of logarithmic $$\mathfrak {sl}_2$$ sl 2 -connections on a curve $${\mathsf {X}}$$ X with fixed generic residues and a category of abelian logarithmic connections on an appropriate spectral double cover "Equation missing". The proof is by constructing a pair of inverse functors $$\pi ^\text {ab}, \pi _\text {ab}$$ π ab , π ab , and the key is the construction of a certain canonical cocycle valued in the automorphisms of the direct image functor $$\pi _*$$ π ∗ .

We prove an existence and uniqueness theorem for exact WKB solutions of general singularly perturbed linear second-order ODEs in the complex domain. These include the one-dimensional time-independent complex Schrödinger equation. Notably, our results are valid both in the case of generic WKB trajectories as well as closed WKB trajectories. We also explain in what sense exact and formal WKB solutions form a basis. As a corollary of the proof, we establish the Borel summability of formal WKB solutions for a large class of problems, and derive an explicit formula for the Borel transform.