Optimality of increasing stability for an inverse boundary value problem

Pu-Zhao Kow, Gunther Uhlmann, Jenn-Nan Wang

Published 2021-02-23Version 1

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.