Uniform WKB, multi-instantons, and resurgent trans-series

2014/01/31 by Gerald V. Dunne, Mithat Ünsal, Mithat Unsal · 6 citations
Mathematics · Physics and Astronomy · #Degenerate energy levels #Eigenvalues and eigenvectors #Instanton #Mathematical analysis #Mathematical physics #Mathematics #Maxima and minima #Physics #Quantum #Quantum Mechanics and Applications #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Quantum mechanics #Realization (probability) #Series (stratigraphy) #WKB approximation #cond-mat.other #hep-th #math-ph #math.MP #quant-ph

paper · pdf · doi:10.1103/physrevd.89.105009

published as Phys. Rev. D 89, 105009 (2014) · 47 pp, 5 figures; v2 as in PRD: typos and figure 5 corrected

openalex publication_date 2014/05/12 · arxiv created 2014/06/20 · arxiv updated 2014/08/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We illustrate the physical significance and mathematical origin of resurgent trans-series expansions for energy eigenvalues in quantum mechanical problems with degenerate harmonic minima, by using the uniform WKB approach. We provide evidence that the perturbative expansion, combined with a global eigenvalue condition, contains all information needed to generate all orders of the nonperturbative multi-instanton expansion. This provides a dramatic realization of the concept of resurgence, whose structure is naturally encoded in the resurgence triangle. We explain the relation between the uniform WKB approach, multi-instantons, and resurgence theory. The essential idea applies to any perturbative expansion, and so is also relevant for quantum field theories with degenerate minima which can be continuously connected to quantum mechanical systems.

Citations

• Lectures on non‐perturbative effects in large N gauge theories, matrix models and strings
• Duality Constraints on String Theory: Instantons and spectral networks

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