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arXiv · 2602.18387

Limiting Absorption Principle and Radiation Condition for the Fractional Helmholtz Equation

Dana Zilberberg·Fioralba Cakoni·Michael S. Vogelius
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Abstract

We investigate elliptic fractional equations in the whole space, involving zero order perturbations of the fractional Laplacian $(-\Delta)^s$, $0 0$, 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.

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Dana Zilberberg, Fioralba Cakoni, Michael S. Vogelius. 2026-02-20. Limiting Absorption Principle and Radiation Condition for the Fractional Helmholtz Equation. https://arxiv.org/abs/2602.18387

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