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arXiv · 2604.12997

Uniqueness and non-uniqueness pairs for the fractional Laplacian

Ricardo Motta
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Abstract

We study uniqueness and non-uniqueness pairs for the fractional Laplacian in the discrete setting. More precisely, under mild regularity assumptions on a function $f$ satisfying $f=0$ on $\Lambda$ and $(-\Delta)^s f=0$ on $M$, where $\Lambda, M \subset \mathbb{R}^d$ are discrete, we find sufficient conditions on these sets that force $f$ to vanish identically, and we provide examples in which non-uniqueness occurs. The uniqueness proofs combine decay estimates forced by discrete zero sets with analytic continuation, while the non-uniqueness results are obtained from an interpolation theorem.

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Ricardo Motta. 2026-04-14. Uniqueness and non-uniqueness pairs for the fractional Laplacian. https://arxiv.org/abs/2604.12997

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