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

Lotfi Hermi, Naoki Saito

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.

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

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.
Paper Structure (31 sections, 10 theorems, 229 equations, 2 figures, 4 tables)

This paper contains 31 sections, 10 theorems, 229 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. From the Integral Operator to the Differential Equation
  3. From the Resolvent to the Integral Operator
  4. Review of resolvents and Green's functions
  5. Our one-dimensional setting
  6. The Nonlocal Boundary Conditions
  7. Kreı̆n-von Neumann Boundary Conditions
  8. Dirichlet Boundary Conditions
  9. Neumann Boundary Conditions
  10. Robin Boundary Conditions
  11. Periodic Boundary Conditions
  12. Anti-Periodic Boundary Conditions
  13. Radoux Boundary Conditions
  14. GSARC Boundary Conditions with Non-Zero Discriminant
  15. GSARC Boundary Conditions with Zero Discriminant
  16. ...and 16 more sections

Key Result

Theorem 4.2

The Dirichlet operator $\mathcal{K_{G}}$ is a rank 2 perturbation of $\mathcal{K}$. Moreover, where $u_1(x), u_2(x)$ are defined in eq:pDevs.

Figures (2)

  • Figure 1: Discriminant condition $\delta - \alpha - 2 \beta + \beta^2 + \alpha \, \delta = 0$ corresponding to $\lambda=0$ being an eigenvalue, with various BCs treated, including the discriminant condition for the "separated" BCs case ($\beta=0$) indicated by the curve on the two sheets of the surface.
  • Figure 2: Examples of GSARS cases treated with respect to the discriminant condition $\cos (\theta_0 - \theta_1) + \cos (\theta_0+\theta_1) - 2 \sin (\theta_0 - \theta_1)=0$, for $\theta_0, \theta_1 \in [0, \pi)$.

Theorems & Definitions (34)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Example 3.5
  • ...and 24 more