Stability estimates for an inverse problem for the Schr\"odinger equation at negative energy in two dimensions

TL;DR

This paper derives three new stability estimates for an inverse Schrödinger problem in two dimensions at negative energy, showing how stability improves with potential smoothness and energy magnitude, reducing ill-posedness.

Contribution

It introduces three novel stability estimates that relate the problem's stability to potential smoothness and energy level, highlighting improved stability at higher energies.

Findings

Stability increases exponentially with potential smoothness.

High energies lead to a transition from logarithmic to Lipschitz stability.

Ill-posedness decreases as the energy magnitude increases.

Abstract

We study the inverse problem of determining a real-valued potential in the two-dimensional Schr\"odinger equation at negative energy from the Dirichlet-to-Neumann map. It is known that the problem is ill-posed and a stability estimate of logarithmic type holds. In this paper we prove three new stability estimates. The main feature of the first one is that the stability increases exponentially with respect to the smoothness of the potential, in a sense to be made precise. The others show how the first estimate depends on the energy, for low and high energies (in modulus). In particular it is found that for high energies the stability estimate changes, in some sense, from logarithmic type to Lipschitz type: in this sense the ill-posedness of the problem decreases when increasing the energy (in modulus).