Stable factorization of the Calderón problem via the Born approximation
Thierry Daudé, Fabricio Macià, Cristóbal J. Meroño, François Nicoleau
TL;DR
This work develops a rigorous Born-level factorization of the radial Calderón problem for Schrödinger operators on the unit ball, proving the existence of a Born approximation $V^{\mathrm{B}}$ and showing that the nonlinear map $V \mapsto V^{\mathrm{B}}$ is locally well-posed and Hölder stable. It links the Calderón problem to one-dimensional inverse spectral theory through Simon's $A$-amplitude, derives an explicit reconstruction algorithm for $V$ from $V^{\mathrm{B}}$, and provides stability and regularization results, including a Fourier-based formula for the regularized Born approximation $V^{\mathrm{B}}_r$. The results yield a clear separation between the ill-posed linear step and a more stable nonlinear recovery, and they obtain a partial DtN characterization for radial potentials via Hausdorff moments. Explicit examples illustrate singularity propagation in the Born approximation and monotonicity properties, while the framework avoids CGO constructions and relies on spectral-theoretic tools. Overall, the paper advances a mathematically rigorous, IST-based approach to stable, layer-stripping-type reconstruction in the radial Calderón setting with practical reconstruction algorithms.
Abstract
In this article we prove the existence of the Born approximation in the context of the radial Calderón problem for Schrödinger operators. The Born approximation naturally appears as the linear component of a factorization of the Calderón problem; we show that the non-linear part, obtaining the potential from the Born approximation, enjoys several interesting properties. First, this map is local, in the sense that knowledge of the Born approximation in a neighborhood of the boundary is equivalent to knowledge of the potential in the same neighborhood, and, second, it is Hölder stable. This proves that the ill-posedness of the Calderón problem arises from the linear step, which consists in computing the Born approximation from the DtN map by solving a Hausdorff moment problem. Moreover, we present an effective algorithm to compute the potential from the Born approximation. Finally, we use the Born approximation to obtain a partial characterization of the set of DtN maps for radial potentials. The proofs of these results do not make use of Complex Geometric Optics solutions or its analogues; they are based on results on inverse spectral theory for Schrödinger operators on the half-line, in particular on the concept of $A$-amplitude introduced by Barry Simon.