Exponential instability in the Gel'fand inverse problem on the energy intervals

TL;DR

This paper demonstrates that the Gel'fand inverse problem exhibits exponential instability even when the Dirichlet-to-Neumann map is restricted to energy intervals, confirming the optimality of existing logarithmic stability estimates.

Contribution

It extends previous instability results to energy interval cases, showing the persistence of exponential instability in the Gel'fand inverse problem.

Findings

Instability remains exponential on energy intervals

Logarithmic stability estimates are proven to be optimal

Extends Mandache's instability results to new settings

Abstract

We consider the Gel'fand inverse problem and continue studies of [Mandache,2001]. We show that the Mandache-type instability remains valid even in the case of Dirichlet-to-Neumann map given on the energy intervals. These instability results show, in particular, that the logarithmic stability estimates of [Alessandrini,1988], [Novikov,Santacesaria,2010] and especially of [Novikov,2010] are optimal (up to the value of the exponent).