Carleman estimate for infinite cylindrical quantum domains and application to inverse problems

TL;DR

This paper develops a Carleman estimate for the Schrödinger equation in infinite cylindrical domains and applies it to achieve Lipschitz stable recovery of the potential from boundary measurements, advancing inverse problem techniques in unbounded domains.

Contribution

It introduces a new Carleman estimate tailored for unbounded cylindrical domains and demonstrates its use in stable inverse potential reconstruction from boundary data.

Findings

Lipschitz stability in potential recovery

Carleman estimate for unbounded domains

Boundary measurement-based inverse solution

Abstract

We consider the inverse problem of determining the time independent scalar potential of the dynamic Schr\"odinger equation in an infinite cylindrical domain, from one Neumann boundary observation of the solution. Assuming that this potential is known outside some fixed compact subset of the waveguide, we prove that it may be Lipschitz stably retrieved by choosing the Dirichlet boundary condition of the system suitably. Since the proof is by means of a global Carleman estimate designed specifically for the Schr\"odinger operator acting in an unbounded cylindrical domain, the Neumann data is measured on an infinitely extended subboundary of the cylinder.