Stable factorization of the Calderón problem via the Born approximation

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 Geometrical Optics solutions or their analogs; they are based on results from inverse spectral theory for Schrödinger operators on the half-line, in particular on the concept of $A$-amplitude introduced by Barry Simon.