A primer of the complex WKB method, with application to the ODE/IM correspondence (2501.05957v2)

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.

• The paper provides a primer on the complex WKB method and applies it to analyze asymptotic properties and spectral determinants of linear ODE solutions.
• Complex WKB is applied to analyze asymptotic behavior near critical points and understand the Stokes phenomenon, crucial for solution structure.
• Complex WKB links Fock-Goncharov coordinates to spectral determinants, enabling asymptotic analysis relevant to the ODE/IM correspondence.

Overview of "A Primer of the Complex WKB Method with Application to the ODE/IM Correspondence"

The paper "A primer of the complex WKB method, with application to the ODE/IM correspondence" by Gabriele Degano and Davide Masoero offers a comprehensive introduction to the complex WKB (Wentzel–Kramers–Brillouin) method. The authors use this method as a tool to derive and prove asymptotic properties of solutions to linear differential equations, particularly focusing on a class of anharmonic oscillators within the context of the ODE/IM correspondence—a linkage between ordinary differential equations and integrable models in mathematical physics.

Lecture Series Structure

The paper is structured as a series of lectures, each addressing a particular aspect of the complex WKB method and its application:

1. Lecture I: Fundamental Theorem of the WKB Approximation - This lecture introduces the fundamental theorem that provides conditions under which a solution to a Schrödinger-type equation can be approximated by a WKB ansatz along a curve. It establishes the theoretical framework necessary for analyzing the differential equation's solutions in the complex plane.
2. Lecture II: Asymptotic Behavior and Stokes Phenomenon - Here, the authors explore the asymptotic behavior of solutions as they approach critical points like zero and infinity—regions where the solution behavior changes markedly. The phenomenon of Stokes lines, critical to understanding how solutions transition between asymptotic regimes, is explained.
3. Lecture III: WKB Theory of Fock-Goncharov Coordinates - This section explores how WKB approximations can be used to calculate Fock-Goncharov coordinates, quantities that encapsulate the global asymptotic behavior of solutions. These concepts allow a reformulation of the spectral problem into a form amenable to WKB analysis.
4. Lecture IV: Asymptotic Analysis of Spectral Determinants - The final lecture focuses on the implications of the WKB method for spectral problems, particularly the Bohr-Sommerfeld quantization conditions. The authors derive asymptotic relations for eigenvalues of the anharmonic oscillator in various limits, providing mathematical rigor to previously heuristic methods.

Key Findings and Implications

• Asymptotic Behavior: The paper provides detailed asymptotic expansions in regimes where traditional methods fail, revealing how solutions to the anharmonic Schrödinger equation behave near singular points.
• Stokes Phenomenon: The transition of wave functions across Stokes lines is crucial for understanding the full analytic structure of solutions, and the authors offer a robust framework for analyzing this behavior.
• Spectral Determinants: By linking Fock-Goncharov coordinates with spectral determinants, new pathways are opened up for the asymptotic paper of eigenvalues, which is a central theme in quantum mechanics and related fields.
• ODE/IM Correspondence: The application of these methods to the ODE/IM correspondence highlights their relevance in linking differential equations with integrable systems, a fundamental area in theoretical physics.

Future Directions

The paper suggests that the complex WKB method's framework could be extended to more complicated systems and potentially other types of differential equations beyond the field initially considered. Moreover, its linkage with Fock-Goncharov coordinates could inspire new analytical techniques for computational physics models.

This foundational work not only contributes to a deeper understanding of the complex WKB method but also lays the groundwork for future explorations into the ODE/IM correspondence, pushing the boundaries of mathematical physics.