Determination of non-compactly supported electromagnetic potentials in unbounded closed waveguide

TL;DR

This paper addresses the inverse problem of uniquely determining electromagnetic potentials in an unbounded waveguide using boundary measurements, even with partial data, by employing complex geometric optics solutions and Carleman estimates.

Contribution

It proves unique recovery of electromagnetic potentials in unbounded waveguides with minimal boundary data assumptions, extending previous results to non-compactly supported potentials.

Findings

Unique recovery of magnetic field and electric potential in unbounded waveguides.

Recovery from boundary measurements restricted to a bounded subset.

Recovery from partial boundary data in cylindrical waveguides.

Abstract

We study the inverse problem of determining a magnetic Schr\"odinger operator in an unbounded closed waveguide from boundary measurements. We consider this problem with a general closed waveguide in the sense that we only require our unbounded domain to be contained into an infinite cylinder. In this context we prove the unique recovery of the magnetic field and the electric potential associated with general bounded and non-compactly supported electromagnetic potentials. By assuming that the electromagnetic potentials are known on the neighborhood of the boundary outside a compact set, we even prove the unique determination of the magnetic field and the electric potential from measurements restricted to a bounded subset of the infinite boundary. Finally, in the case of a waveguide taking the form of an infinite cylindrical domain, we prove the recovery of the magnetic field and the electric potential from partial data corresponding to restriction of Neumann boundary measurements to slightly more than half of the boundary. We establish all these results by mean of suitable complex geometric optics solutions and Carleman estimates suitably designed for our problem stated in an unbounded domain and with bounded electromagnetic potentials.