## 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 language | English (US) |
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Pages (from-to) | 59-83 |

Number of pages | 25 |

Journal | Applied and Computational Harmonic Analysis |

Volume | 45 |

Issue number | 1 |

DOIs | |

State | Published - Jul 2018 |

## Keywords

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

## ASJC Scopus subject areas

- Applied Mathematics