On Rayleigh-type formulas for a non-local boundary value problem associated with an integral operator commuting with the Laplacian. (English) Zbl 1443.47047
Summary: 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 \(| x - y |^\rho\), \(0 < \rho \leq 1\), \(x, y \in [-a, a]\). 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 the second author [Appl. Comput. Harmon. Anal. 25, No. 1, 68–97 (2008; Zbl 1148.94005)]. We also discuss extensions in higher dimensions and links with distance matrices.
MSC:
| 47G10 | Integral operators |
| 47A75 | Eigenvalue problems for linear operators |
| 45P05 | Integral operators |
| 65J10 | Numerical solutions to equations with linear operators |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
Keywords:
Rayleigh function; eigenvalues; non-local boundary value problem; integral operator; LaplacianCitations:
Zbl 1148.94005Software:
DLMFReferences:
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