Simultaneous reconstruction of two potentials for a nonconservative Schr\"odinger equation with dynamic boundary conditions

TL;DR

This paper addresses an inverse problem for a Schrödinger equation with mixed boundary conditions, establishing Lipschitz stability for reconstructing two potentials from a single flux measurement using Carleman estimates.

Contribution

It introduces a novel stability estimate for simultaneously reconstructing two potentials in a nonconservative Schrödinger equation with dynamic boundary conditions.

Findings

Lipschitz stability estimate derived

Single measurement suffices for reconstruction

Utilizes Bukhgeim-Klibanov method with Minkowski-based Carleman estimate

Abstract

In this article, we consider an inverse problem involving the simultaneous reconstruction of two real valued potentials for a Schr\"odinger equation with mixed boundary conditions: a dynamic boundary condition of Wentzell type and a Dirichler boundary condition. The main result of this paper is a Lipschitz stability estimate for such potentials from a single measurement of the flux. This result is deduced using the Bukhgeim-Klibanov method and a suitable Carleman estimate where the weight function depends on Minkowski's functional.