An inverse stability result for non compactly supported potentials by one arbitrary lateral Neumann observation

TL;DR

This paper proves that the scalar potential in an infinite cylindrical Schrödinger equation can be stably reconstructed from a single boundary observation, even when the potential is only known near the boundary, advancing inverse problem theory.

Contribution

It establishes a logarithmic stability result for inverse Schrödinger problems with partial boundary data in infinite waveguides, even with minimal potential knowledge.

Findings

Potential can be uniquely determined from boundary measurements.

Stability estimate is logarithmic, indicating robustness.

Applicable to arbitrary boundary strips, broadening inverse problem scope.

Abstract

In this paper we investigate the inverse problem of determining the time independent scalar potential of the dynamic Schr\"odinger equation in an infinite cylindrical domain, from partial measurement of the solution on the boundary. Namely, if the potential is known in a neighborhood of the boundary of the spatial domain, we prove that it can be logarithmic stably determined in the whole waveguide from a single observation of the solution on any arbitrary strip-shaped subset of the boundary.