A primer of the complex WKB method, with application to the ODE/IM correspondence

TL;DR

This paper introduces the complex WKB method through anharmonic oscillators, exploring its key concepts and applying it to analyze spectral asymptotics in the ODE/IM correspondence.

Contribution

It provides a comprehensive introduction to the complex WKB method and applies it to compute spectral asymptotics for anharmonic oscillators within the ODE/IM framework.

Findings

Derived asymptotic behavior of the spectrum for fixed momentum and large energy.

Proved spectral asymptotics when both momentum and energy are large.

Clarified the role of Stokes phenomena and quadratic differentials in spectral analysis.

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.