Determining a Magnetic Schrödinger Operator from Partial Cauchy Data

Abstract

Abstract. In this paper we show, in dimension n ≥ 3, that knowledge of the Cauchy data for the Schrödinger equation in the presence of a magnetic potential, measured on possibly very small subsets of the boundary, determines uniquely the magnetic field and the electric potential. We follow the general strategy of [7] using a richer set of solutions to the Dirichlet problem that has been used in previous works on this problem.

The Calderón problem with partial data in two dimensions

“…The method of the proof uses Carleman estimates with nonlinear weights. The case of the magnetic Schrödinger equation was considered in [11] and an improvement on the regularity of the coefficients is done in [21]. Stability estimates for the magnetic Schrödinger equation with partial data were proven in [30].…”

Semiclassical Pseudodifferential Calculus and the Reconstruction of a Magnetic Field

“…All these results are nonconstructive. We also mention the recent papers [2], [6] and [28] which consider boundary determination, partial Cauchy data and stability for this inverse problem. In this paper we give a constructive algorithm for recovering curl W and q from Λ W,q .…”

On the linearized local Calderón problem

“…Other applications involve the proof of Helgason's support theorem on the Radon transform and extensions (see [4] and [11]) of this result. Theorem 1.3 has also proved to be a useful tool in the resolution of inverse problems (see [16] and [9]) with partial data. In fact the microlocal version of Holmgren's uniqueness theorem is a consequence 1 of a more general result on the analytic wave front set due to Kashiwara (see [15], [23] chapter 8, Theorem 8.3, [10] chapter 9, Theorem 9.6.6) Watermelon Theorem.…”