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

TL;DR

This paper establishes Lipschitz stability estimates for the finite dimensional fractional Calderón problem using finitely many measurements, under the assumption that the potential lies in a finite-dimensional subspace, leveraging the strong Runge approximation property.

Contribution

It proves Lipschitz stability for the fractional Calderón problem with finite data when the potential is in a finite-dimensional subspace, extending previous stability results.

Findings

Lipschitz stability estimates are obtained for the problem.

Finite Cauchy data suffice for stable reconstruction.

The approach accommodates zero as a Dirichlet eigenvalue.

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.