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

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 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.

Minor changes: fixed some typos, updated references. arXiv admin note: text overlap with arXiv:2008.06492

References in corpus (2)

• Exact WKB methods in $SU(2)$ $N_f=1$
• The complex geometry of the free particle, and its perturbations

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