What question did this study set out to answer?

The aim is to explore the spectral characteristics and numerical behavior of the discrete Abel integral operator.

May 6, 2026Open Access

On the Spectral Norm of the Discrete Abel Integral Operator: Numerical Convergence and the Emergence of the Ontological Conductivity Constant C∞

Key Points

The aim is to explore the spectral characteristics and numerical behavior of the discrete Abel integral operator.
Investigated the spectral properties of the symmetric Abel integral operator on L2[0, 1] using numerical methods.
Utilized high-precision numerical discretization and finite-size scaling.
Analyzed the convergence of the operator norm for an N × N quadrature discretization.
Identified a fundamental constant C∞ approximately equal to 2.6828.
Demonstrated operator norm converges according to the relation ∥TN∥ ∼ C∞√N.
Proposed the fundamental constant as an invariant relating to embedded field structures.

Abstract

AbstractWe investigate the spectral properties of the symmetric Abel integral operatorT on L20, 1 with the kernel K(x, y) = |x − y|−1/2. While the tail of the spectrumis well-described by the Dostani´c asymptotic, the operator norm ∥T ∥ (the principaleigenvalue) lacks a closed-form expression. Through high-precision numericaldiscretization and finite-size scaling, we establish the existence of a fundamentalconstant C∞ ≈ 2.6828. We demonstrate that the operator norm for an N × Nquadrature discretization satisfies ∥TN∥ ∼ C∞√N. This constant is proposed as astructural invariant of the “OntoField,” representing the maximum resonant capacityof a localized field segment.

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