An inverse problem for the magnetic Schr\"{o}dinger operator on Riemannian manifolds from partial boundary data

TL;DR

This paper addresses the inverse problem of determining magnetic and potential terms in a Schrödinger operator on Riemannian manifolds using partial boundary data, employing Carleman estimates and geometric optics solutions.

Contribution

It introduces a method to recover magnetic and potential terms from partial boundary measurements on Riemannian manifolds, advancing inverse problem techniques.

Findings

Proved uniqueness of the inverse problem under certain conditions.

Developed Carleman estimates for functions vanishing on part of the boundary.

Constructed complex geometric optics solutions for partial boundary data.

Abstract

We consider the inverse problem of recovering the magnetic and potential term of a magnetic Schr\"{o}dinger operator on certain compact Riemannian manifolds with boundary from partial Dirichlet and Neumann data on suitable subsets of the boundary. The uniqueness proof relies on proving a suitable Carleman estimate for functions which vanish only on a part of boundary and constructing complex geometric optics solutions which vanish on a part of the boundary.