A Holder-logarithmic stability estimate for an inverse problem in two dimensions

TL;DR

This paper establishes a new stability estimate for recovering potentials in a 2D Schrödinger equation, showing increased stability at high energies and linking stability to potential regularity.

Contribution

It introduces an explicit stability estimate that improves with energy, demonstrating a transition from logarithmic to Hölder stability in inverse problems.

Findings

Stability improves at high energies.

Potential regularity influences stability.

Develops estimates for a non-local Riemann-Hilbert problem.

Abstract

The problem of the recovery of a real-valued potential in the two-dimensional Schrodinger equation at positive energy from the Dirichlet-to-Neumann map is considered. It is know that this problem is severely ill-posed and the reconstruction of the potential is only logarithmic stable in general. In this paper a new stability estimate is proved, which is explicitly dependent on the regularity of the potentials and on the energy. Its main feature is an efficient increasing stability phenomenon at sufficiently high energies: in some sense, the stability rapidly changes from logarithmic type to Holder type. The paper develops also several estimates for a non-local Riemann-Hilbert problem which could be of independent interest.