# Stable determination of a vector field in a non-self-adjoint dynamical Schr\"odinger equation on Riemannian manifolds

## TL;DR

This paper establishes a stable method to determine a vector field in a non-self-adjoint Schrödinger equation on Riemannian manifolds using Carleman estimates, advancing inverse problem techniques in geometric analysis.

## Contribution

It introduces a new stability estimate for the inverse problem of recovering a vector field on Riemannian manifolds, extending previous results to non-self-adjoint cases.

## Key findings

- Proves Hölder stability estimate for the inverse problem in dimension n > 2.
- Reduces the problem to an electromagnetic Schrödinger equation.
- Utilizes a specialized Carleman estimate for elliptic operators.

## Abstract

This paper deals with an inverse problem for a non-self-adjoint Schr\"odinger equation on a compact Riemannian manifold. Our goal is to stably determine a real vector field from the dynamical Dirichlet-to Neumann map. We establish in dimension n greater than 2, an H\"older type stability estimate for the inverse problem under study. The proof is mainly based on the reduction to an equivalent problem for an electro-magnetic Schr\"odinger equation and the use of a Carleman estimate designed for elliptic operators.