On Rayleigh-Type Formulas for a Non-local Boundary Value Problem Associated with an Integral Operator Commuting with the Laplacian

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 $|x-y{|}^{\rho }$, $0<\rho \le 1$, $x,y\in [-a,a]$. 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 $\rho =1$, 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.

24 pages, 1 figure

References in corpus (3)

• Spectral properties of distance matrices
• The one-dimensional Schrödinger-Newton equations
• Distance matrices and isometric embeddings