Existence and Uniqueness of Exact WKB Solutions for Second-Order Singularly Perturbed Linear ODEs

TL;DR

This paper establishes the existence, uniqueness, and Borel summability of exact WKB solutions for second-order singularly perturbed linear ODEs, including complex Schrödinger equations, in the complex domain.

Contribution

It provides a rigorous proof of existence and uniqueness of exact WKB solutions and clarifies their basis properties, extending previous results to more general trajectories.

Findings

Proves existence and uniqueness of exact WKB solutions.

Shows Borel summability of formal WKB solutions.

Provides explicit Borel transform formula.

Abstract

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 Schr\"odinger 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.