Optimality of increasing stability for an inverse boundary value problem

TL;DR

This paper investigates the stability properties of an inverse boundary value problem for the Schrödinger equation, demonstrating that the transition from exponential to Hölder stability with increasing frequency is optimal.

Contribution

It proves that the change from exponential to Hölder stability as frequency increases is the best possible, confirming the optimality of previous results.

Findings

Stability improves from exponential to Hölder with increasing frequency.

The results confirm the optimality of previous stability estimates.

Provides rigorous proof of the limits of stability improvements.

Abstract

In this work we study the optimality of increasing stability of the inverse boundary value problem (IBVP) for Schr\"{o}dinger equation. The rigorous justification of increasing stability for the IBVP for Schr\"{o}dinger equation were established by Isakov \cite{Isa11} and by Isakov, Nagayasu, Uhlmann, Wang of the paper \cite{INUW14}. In \cite{Isa11}, \cite{INUW14}, the authors showed that the stability of this IBVP increases as the frequency increases in the sense that the stability estimate changes from a logarithmic type to a H\"{o}lder type. In this work, we prove that the instability changes from an exponential type to a H\"older type when the frequency increases. This result verifies that results in \cite{Isa11}, \cite{INUW14} are optimal.