A local uniqueness theorem for the fractional Schr\"{o}dinger equation on closed Riemannian manifolds

TL;DR

This paper proves a local uniqueness theorem for recovering the potential in the fractional Schrödinger equation on closed Riemannian manifolds, utilizing a new Runge approximation property.

Contribution

It introduces a novel local uniqueness result for the fractional Schrödinger equation on manifolds and establishes a new Runge approximation property for this setting.

Findings

Potential V can be recovered locally from the source-to-solution map.

A new Runge approximation property is derived for the fractional Schrödinger equation.

The method applies to smooth connected closed Riemannian manifolds.

Abstract

We investigate that a potential $V$ in the fractional Schr\"odinger equation $( (-\Delta_g )^s +V ) u=f$ can be recovered locally by using the local source-to-solution map on smooth connected closed Riemannian manifolds. To achieve this goal, we derive a related new Runge approximation property.