A Closed-Form Approach to Oscillatory Integrals in Level-Crossing Physics

2025/06/29 by M. B. Kenmoe, Kenmoe, Maseim B., A.D. Kammogne +1
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #Model Reduction and Neural Networks #Numerical methods for differential equations #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2506.23385

openalex publication_date 2025/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present a closed-form, exact analytical solution, valid at finite times, to a class of multiple integrals with highly oscillatory kernels. Our approach leverages the intimate connection between these integrals and the minimal level-crossing model, namely the Landau-Zener model. Benchmarking against data from numerical simulations demonstrates excellent agreement validating our analytical method. Impacts of our results in level-crossing dynamics are also discussed. A dedicated Mathematica package named \bf OscillatoryIntegralAnalytical.wl publicly accessible in our \hrefhttps://github.com/Kenmax15/Closed-Form-Approach-to-Oscillatory-Integrals/tree/mainGitHub repository allows generating the integrals for arbitrary order.

Citations

• When Mathematics Helps Physics: Calculation of the Integral of Kholodenko and Silagadze
• Efficient evaluation of real-time path integrals
• On the adaptive Levin method
• Oscillatory path integrals for radio astronomy
• Cancellations in power series of sine type
• Yet Another Table of Integrals
• Matrix exponential solution of the Landau-Zener problem
• Atomi orientati in campo magnetico variabile

Related

• A method for bounding oscillatory integrals in terms of non-oscillatory integrals
• Derivation of the Landau-Zener formula via functional equations
• An Approach To Endpoint Problems in Oscillatory Singular Integrals
• On the generalized Landau-Zener problem
• Resolution of singularities, asymptotic expansions of oscillatory integrals, and related phenomena