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arXiv · 1805.00866

Lipschitz stability for the Finite Dimensional Fractional Calder\'on Problem with Finite Cauchy Data

Angkana Rüland·Eva Sincich
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Abstract

In this note we discuss the conditional stability issue for the finite dimensional Calder\'on problem for the fractional Schr\"{o}dinger equation with a finite number of measurements. More precisely, we assume that the unknown potential $q \in L^{\infty}(\Omega) $ in the equation $((-\Delta)^s+ q)u = 0 \mbox{ in } \Omega\subset \mathbb{R}^n$ satisfies the a priori assumption that it is contained in a finite dimensional subspace of $L^{\infty}(\Omega)$. Under this condition we prove Lipschitz stability estimates for the fractional Calder\'on problem by means of finitely many Cauchy data depending on $q$. We allow for the possibility of zero being a Dirichlet eigenvalue of the associated fractional Schr\"odinger equation. Our result relies on the strong Runge approximation property of the fractional Schr\"odinger equation.

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Angkana Rüland, Eva Sincich. 2018-05-02. Lipschitz stability for the Finite Dimensional Fractional Calder\'on Problem with Finite Cauchy Data. https://arxiv.org/abs/1805.00866

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