On Rayleigh-type formulas for a non-local boundary value problem associated with an integral operator commuting with the Laplacian (with L. Hermi), Applied and Computational Harmonic Analysis, vol.45, no.1, pp.59-83, 2018.


In this article we prove the existence, uniqueness, and simplicity of a negative eigenvalue for a class of integral operators whose kernel is of the form |x-y|ρ, 0 < ρ ≤ 1, x, y ∈ [-a, a]. We also provide two different ways of producing recursive formulas for the Rayleigh functions (i.e., recursion formulas for power sums) of the eigenvalues of this integral operator when ρ=1, providing means of approximating this negative eigenvalue. These methods offer recursive procedures for dealing with the eigenvalues of a one-dimensional Laplacian with non-local boundary conditions which commutes with an integral operator having a harmonic kernel. The problem emerged in recent work by one of the authors [45]. We also discuss extensions in higher dimensions and links with distance matrices.

Keywords: Rayleigh functions, Laplacian eigenvalue problems, non-local boundary conditions, sum rules, power sums, Euler-Rayleigh method, distance matrices

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