On a Nonlocal Integral Operator Commuting with the Laplacian and the Sturm-Liouville Problem I: Low Rank Perturbations of the Operator

TL;DR

This paper demonstrates that all real coupled self-adjoint boundary value problems can be expressed as finite rank polynomial perturbations of the free space Green's function, capturing boundary conditions in a fundamental way.

Contribution

It reformulates boundary value problems as integral operators and shows they are finite rank polynomial perturbations of the free space Green's function.

Findings

Perturbations are polynomials of rank up to 4.

Boundary conditions are encapsulated by these perturbations.

All problems can be represented as integral operators with finite rank modifications.

Abstract

We reformulate all general real coupled self-adjoint boundary value problems as integral operators and show that they are all finite rank perturbations of the free space Green's function on the real line. This free space Green's function corresponds to the nonlocal boundary value problem proposed earlier by Saito [N. Saito, Appl. Comput. Harmonic Anal., 25, 68--97 (2008)]. We prove these perturbations to be polynomials of rank up to 4. They encapsulate in a fundamental way the corresponding boundary conditions.