Stability estimate in an inverse problem for non-autonomous Schr\"odinger equations

TL;DR

This paper establishes Lipschitz stability estimates for recovering a time-dependent magnetic field in a Schrödinger equation from boundary measurements, advancing inverse problem theory for quantum systems.

Contribution

It provides the first Lipschitz stability result for the inverse problem of determining a non-autonomous magnetic potential in Schrödinger equations using finite boundary data.

Findings

Lipschitz stability of the magnetic potential is proven.

Stability depends on changing initial values multiple times.

Results apply to Schrödinger equations in bounded domains.

Abstract

We consider the inverse problem of determining the time dependent magnetic field of the Schr\"odinger equation in a bounded open subset of $R^n$, with $n \geq 1$, from a finite number of Neumann data, when the boundary measurement is taken on an appropriate open subset of the boundary. We prove the Lispchitz stability of the magnetic potential in the Coulomb gauge class by $n$ times changing initial value suitably.