Stable determination of coefficients in the dynamical Schrödinger equation in a magnetic field

Abstract

ABSTRACT. In this paper we consider the inverse problem of determining on a compact Riemannian manifold the electric potential or the magnetic field in a Schrödinger equation with Dirichlet data from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichlet-to-Neumann map associated to the magnetic Schrödinger equation. We prove in dimension n ě 2 that the knowledge of the Dirichlet-to-Neumann map for the Schrödinger equation uniquely determines the magnetic field and the electric potential and we establish Hölder-type stability.

Stability estimate for an inverse problem for the Schrödinger equation in a magnetic field with time-dependent coefficient

“…In the Riemannian case, Bellassoued [2] proved recently a Hölder-type stability estimate in the recovery of the magnetic field dα A and the time-independent electric potential q from the knowledge of the Dirichletto-Neumann map associated to the Shrödinger equation with zero initial data. In the absence of the magnetic potential A, the problem of recovering the electric potential q on a compact Riemannian manifold was solved by Bellassoued and Dos Santos Ferreira [7].…”

“…In the present paper, we address the uniqueness and the stability issues in the inverse problem of recovering the magnetic field dα A and the time-dependent potential q in the dynamical Schrödinger equation, from the knowledge of the operator Λ A,q . By means of techniques used in [2,9], we prove a "log-type" stability estimate in the recovery of the magnetic field and a "log-log-log-type" stability inequality in the determination of the time-dependent electric potential.…”

Stable determination of a vector field in a non-Self-Adjoint dynamical Schrödinger equation on Riemannian manifolds

“…In order to deal with our main problem we rst need to deal with this equivalent problem: Problem 2: Is it possible to stably recover the electric potential q and the solenoidal part A s of the covector A dened on a simple compact Riemannian manifold from the knowledge of the DN map N A,q under certain conditions? Actually, Problem 2 is closely related to the one considered by Bellassoued [1] in the case where the covector eld A is with real valued coecients. But here we formulate the problem for complex vector elds.…”

“…In this paper, our objective is the study of the inverse problem associated with the equation (2). Inspired by the work of Bellassoued and Rezig [5], Bellassoued and Ben Aïcha [2] and the paper of Bellassoued [1], we aim to stably recover the vector eld X from the DN map Λ X and show a stability of Hölder type. It seems that the present paper is the rst proving a stability result for a Riemannian non-self-adjoint operator.…”

Simultaneous determination of the magnetic field and the electric potential in the Schrödinger equation by a finite number of boundary observations

“…In the Riemannian case, Bellassoued [2] proved recently a Hölder-type stability estimate in the recovery of the magnetic field da and the electric potential q from the knowledge of the Dirichlet-to-Neumann map associated to the Schrödinger equation with zero initial data. In the absence of the magnetic potential a, the problem of recovering the electric potential q on a compact Riemannian manifold was solved by Bellassoued and Dos Santos Ferreira [8].…”