TY - JOUR

T1 - On Rayleigh-type formulas for a non-local boundary value problem associated with an integral operator commuting with the Laplacian

AU - Hermi, Lotfi

AU - Saito, Naoki

PY - 2018/7

Y1 - 2018/7

N2 - 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.

AB - 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.

KW - Eigenvalues

KW - Integral operator

KW - Laplacian

KW - Non-local boundary value problem

KW - Rayleigh function

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U2 - 10.1016/j.acha.2016.08.003

DO - 10.1016/j.acha.2016.08.003

M3 - Article

AN - SCOPUS:85005942706

VL - 45

SP - 59

EP - 83

JO - Applied and Computational Harmonic Analysis

JF - Applied and Computational Harmonic Analysis

SN - 1063-5203

IS - 1

ER -