Inverse problem for fractional Schr\"{o}dinger equations with drift on closed Riemannian manifolds

TL;DR

This paper establishes the unique determination of the Riemannian metric, drift, and potential in fractional Schrödinger equations on closed manifolds from partial boundary data, extending inverse problem results to fractional and geometric settings.

Contribution

It introduces a method to recover metric, drift, and potential simultaneously for fractional Schrödinger equations on closed manifolds, incorporating Runge approximation techniques.

Findings

Unique determination of metric, drift, and potential from partial data

Extension of inverse problems to fractional Schrödinger equations on manifolds

Use of Runge approximation to recover drift term

Abstract

This paper is concerned about the inverse coefficient problems of variable-coefficient fractional Schr\"{o}dinger equations with drift on connected closed Riemannian manifolds. We prove that the knowledge of the underlying equation of order $\alpha\in (\frac{1}{2},1)$ on any non-empty open subset of the underlying manifold determines the Riemannian metric, the drift and the potential, simultaneously and uniquely, up to a gauge transformation, under the same geometric assumptions on the observation set as in \cite{feizmohammadi2024calderonproblemfractionalschrodinger}. The method of proof is based on that of \cite{feizmohammadi2024calderonproblemfractionalschrodinger} for fractional Schr\"{o}dinger operators, with the incorporation of the Runge approximation to recover the drift term.