A primer of the complex WKB method, with application to the ODE/IM correspondence
Abstract
In these lectures, we provide an introduction to the complex WKB method, using as a guiding example a class of anharmonic oscillators that appears in the ODE/IM correspondence. In the first three lectures, we introduce the main objects of the method, such as the WKB function, the integral equations of Volterra type, the quadratic differential and its horizontal/Stokes lines, the Stokes phenomenon, the notion of asymptotic values, the Fock-Goncharov coordinates and their WKB approximation. In the fourth and last lecture, we compute (and prove) the asymptotic behaviour of the spectrum of the anharmonic oscillators in two asymptotic regimes, when the momentum is fixed and the energy is large, and when the momentum (hence also the energy) is large.
Cite
@article{arxiv.2501.05957,
title = {A primer of the complex WKB method, with application to the ODE/IM correspondence},
author = {Gabriele Degano and Davide Masoero},
journal= {arXiv preprint arXiv:2501.05957},
year = {2025}
}
Comments
14 figures, 59 pages. Lectures prepared for the summer school 'Contemporary trends in integrable systems', July 2024, Lisbon. In v2, a few typos were corrected, and a few sentences were reworded