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

Yavar Kian (1, 2)

1 CPT - Centre de Physique Théorique - UMR 7332
2 CPT - E8 Dynamique quantique et analyse spectrale

We study the inverse problem of determining a magnetic Schrödinger 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 unique recovery of general bounded and non-compactly supported electromagnetic potentials modulo gauge invariance. 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 electromagnetic 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 general electromagnetic potentials 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.

Mots clés: Inverse problems; unbounded; domain; elliptic equations; electromagnetic potential; Carleman estimate; unbounded domain; closed waveguide; partial data

Domaines: Equations aux dérivées partielles [math.AP]

Yavar Kian. Determination of non-compactly supported electromagnetic potentials in unbounded closed waveguide. Revista Matemática Iberoamericana, 2020, 36 (3), pp.671-710. ⟨10.4171/rmi/1143⟩. ⟨hal-01707637v2⟩