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Limiting Absorption Principle and Radiation Condition for the Fractional Helmholtz Equation

Dana Zilberberg, Fioralba Cakoni, Michael S. Vogelius

Abstract

We investigate elliptic fractional equations in the whole space, involving zero order perturbations of the fractional Laplacian $(-Δ)^s$, $0<s<1$. Our main objective is to determine appropriate radiation conditions at infinity that ensure existence and uniqueness of solutions to the fractional type Helmholtz equation. Extending classical scattering theory for the Helmholtz equation, we introduce and analyze suitable Sommerfeld type radiation conditions for fractional orders. A central contribution is the explicit computation of the outgoing free space Greens function for the operator $(-Δ)^s-k^{2s}$, for all $0<s<1$, any dimension and $k>0$, obtained via contour integration and a limiting absorption principle. We show that its asymptotic behavior at infinity coincides with a rescaled version of the classical Helmholtz fundamental solution, thereby justifying the standard Sommerfeld radiation condition for compactly supported sources. In addition, using resolvent estimates and a limiting absorption framework, we establish existence and uniqueness of outgoing solutions for compactly supported data, and for weighted sources. We further derive a convolution representation of the solution in terms of the outgoing fundamental solution. For inhomogeneous media with compactly supported perturbations, we reformulate the problem as a Lippmann Schwinger integral equation of Fredholm type and prove unique solvability away from a discrete set of frequencies. Our analysis provides a rigorous foundation for scattering theory of fractional Helmholtz operators and offers a framework suitable for numerical implementation of these nonlocal wave propagation models.

Limiting Absorption Principle and Radiation Condition for the Fractional Helmholtz Equation

Abstract

We investigate elliptic fractional equations in the whole space, involving zero order perturbations of the fractional Laplacian , . Our main objective is to determine appropriate radiation conditions at infinity that ensure existence and uniqueness of solutions to the fractional type Helmholtz equation. Extending classical scattering theory for the Helmholtz equation, we introduce and analyze suitable Sommerfeld type radiation conditions for fractional orders. A central contribution is the explicit computation of the outgoing free space Greens function for the operator , for all , any dimension and , obtained via contour integration and a limiting absorption principle. We show that its asymptotic behavior at infinity coincides with a rescaled version of the classical Helmholtz fundamental solution, thereby justifying the standard Sommerfeld radiation condition for compactly supported sources. In addition, using resolvent estimates and a limiting absorption framework, we establish existence and uniqueness of outgoing solutions for compactly supported data, and for weighted sources. We further derive a convolution representation of the solution in terms of the outgoing fundamental solution. For inhomogeneous media with compactly supported perturbations, we reformulate the problem as a Lippmann Schwinger integral equation of Fredholm type and prove unique solvability away from a discrete set of frequencies. Our analysis provides a rigorous foundation for scattering theory of fractional Helmholtz operators and offers a framework suitable for numerical implementation of these nonlocal wave propagation models.
Paper Structure (26 sections, 35 theorems, 191 equations, 2 figures, 1 table)

This paper contains 26 sections, 35 theorems, 191 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Outgoing Fundamental Solution of TEXT
  3. Computations in 1D for TEXT
  4. Computations in 2D for TEXT
  5. The case of TEXT
  6. TEXT
  7. Computations in 2D for TEXT
  8. Computations in 3D for TEXT
  9. Computations in 3D for TEXT
  10. Asymptotics at infinity of the outgoing fundamental solution
  11. Extended Resolvent of the Fractional Laplacian and the Radiation Condition at Infinity
  12. Properties of the resolvent in weighted Sobolev spaces
  13. Uniqueness given by the generalized Sommerfeld radiation condition
  14. Volume Integral Representation of the Resolvent
  15. The inhomogeneous fractional Helmholtz equation and the radiation condition
  16. ...and 11 more sections

Key Result

Theorem 2.2

The outgoing fundamental solution $G_{n,s}^{k}$ of the fractional Helmholtz operator $(-\Delta)^s - k^{2s}$ for $k>0$ and $0<s<1$ in $\mathbb{R}^n$, $n=1,2,3$, behaves asymptotically as $|x|\to \infty$ like the outgoing fundamental solution of the Helmholtz operator $\Delta + k^2$. In particular, it

Figures (2)

  • Figure 1: Contour integral in the complex plane
  • Figure 2: Sketch of the profile of the cut-off functions and the estimate regions of ${\mathbb C}$

Theorems & Definitions (76)

  • Remark 2.1
  • Theorem 2.2
  • Theorem 3.1: Ben-Artzi, Nemirovski ben1997remarks
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Definition 3.1: GSRC
  • Lemma 3.4
  • proof
  • ...and 66 more