The Sturm-Liouville Hierarchy of evolution equations and the Weyl functions of related spectral problems

2025/09/24 by Paola Rubbioni, Rubbioni, Paola, Anna Rita Sambucını +3
Mathematics · Physics and Astronomy · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2509.20536

openalex publication_date 2025/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The goal of the present paper is that of defining the so-called Sturm-Liouville hierarchy of evolution equations, firstly by using the zero-curvature formalism, then by using the asymptotic properties of the Weyl m-functions for certain classes of pairs (q,y) of the Sturm-Liouville eigenvalue equation -φ''+qφ=λyφ, in the space L²(\mathbb R,ydx). Since the Weyl m-functions are known to contain all the information concerning the spectral properties of the above equation, the determination of the evolution of some spectral characteristics will allow to determine solutions of the hierarchies of evolution equations.

Citations

• Variable depth KDV equations and generalizations to more nonlinear regimes
• On the inverse scattering approach to the Camassa-Holm equation

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• On a spectral theorem of Weyl
• Spectral partitions for Sturm-Liouville problems
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