Existence and Uniqueness of Exact WKB Solutions for Second-Order Singularly Perturbed Linear ODEs
Analysis of PDEs
2023-06-07 v2 Mathematical Physics
Classical Analysis and ODEs
math.MP
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.
Keywords
Cite
@article{arxiv.2106.10248,
title = {Existence and Uniqueness of Exact WKB Solutions for Second-Order Singularly Perturbed Linear ODEs},
author = {Nikita Nikolaev},
journal= {arXiv preprint arXiv:2106.10248},
year = {2023}
}
Comments
Minor changes: fixed some typos, updated references. arXiv admin note: text overlap with arXiv:2008.06492