2015/12/04 by Boaz Haberman · 1 citation
Mathematics · #Ball (mathematics) #Boundary value problem #Bounded function #Magnetic field #Magnetic potential #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Operator (biology) #Potential field #Spectral Theory in Mathematical Physics #Uniqueness #math.AP
paper · pdf · doi:10.1093/imrn/rnw263
published in International Mathematics Research Notices, rnw263 (Oxford University Press)
arxiv created 2015/12/04 · openalex created_date 2016/06/24 · openalex publication_date 2016/10/31 · arxiv updated 2017/03/01 · openalex updated_date 2026/08/05
We consider the Gel’fand–Calderón problem for a Schrödinger operator of the form |-(∇ + iA)2 + q|, defined on a ball |B| in |ℝ3|. We assume that the magnetic potential |A| is small in |Ws,3| for some |s>0|, and that the electric potential |q| is in |W-1,3|. We show that, under these assumptions, the magnetic field |curl A| and the potential |q| are both determined by the Dirichlet–Neumann relation at the boundary |∂ B|. The assumption on |q| is critical with respect to homogeneity, and the assumption on |A| is nearly critical. Previous uniqueness theorems of this type have assumed either that both |A| and |q| are bounded or that |A| is zero.