On Rayleigh-type formulas for a non-local boundary value problem associated with an integral operator commuting with the Laplacian

Lotfi Hermi, Naoki Saito

Research output: Contribution to journalArticle

Abstract

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 [48]. We also discuss extensions in higher dimensions and links with distance matrices.

Original languageEnglish (US)
Pages (from-to)59-83
Number of pages25
JournalApplied and Computational Harmonic Analysis
Volume45
Issue number1
DOIs
StatePublished - Jul 2018

Keywords

  • Eigenvalues
  • Integral operator
  • Laplacian
  • Non-local boundary value problem
  • Rayleigh function

ASJC Scopus subject areas

  • Applied Mathematics

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