Lipschitz stability of an inverse boundary value problem for a Schr\"{o}dinger type equation

TL;DR

This paper establishes a Lipschitz stability estimate for the inverse boundary value problem of determining a piecewise constant potential in a Schrödinger equation from boundary measurements, improving upon the typical logarithmic stability.

Contribution

The work provides the first Lipschitz stability result for the inverse Schrödinger problem under piecewise constant potentials with known bounds and a fixed number of unknowns.

Findings

Lipschitz stability estimate proven for piecewise constant potentials

The stability improves upon the usual logarithmic estimates

Applicable to potentials with a known number of unknown values

Abstract

In this paper we study the inverse boundary value problem of determining the potential in the Schr\"{o}dinger equation from the knowledge of the Dirichlet-to-Neumann map, which is commonly accepted as an ill-posed problem in the sense that, under general settings, the optimal stability estimate is of logarithmic type. In this work, a Lipschitz type stability is established assuming a priori that the potential is piecewise constant with a bounded known number of unknown values.