Stability estimates for partial data inverse problems for Schr\"odinger operators in the high frequency limit

TL;DR

This paper investigates the inverse boundary problem for Schr"odinger operators at a fixed frequency, demonstrating that high-frequency data yields improved stability estimates for potential recovery from partial boundary measurements.

Contribution

It establishes that the stability of potential reconstruction from partial boundary data improves from logarithmic to Hölder type as the frequency increases.

Findings

Logarithmic stability at fixed frequency

Hölder stability in the high frequency regime

Potential determined from partial Robin-to-Dirichlet map

Abstract

We consider the partial data inverse boundary problem for the Schr\"odinger operator at a frequency $k>0$ on a bounded domain in $\mathbb{R}^n$, $n\ge 3$, with impedance boundary conditions. Assuming that the potential is known in a neighborhood of the boundary, we first show that the knowledge of the partial Robin-to-Dirichlet map at the fixed frequency $k>0$ along an arbitrarily small portion of the boundary, determines the potential in a logarithmically stable way. We prove, as the principal result of this work, that the logarithmic stability can be improved to the one of H\"older type in the high frequency regime.