Math. Ann. 303, 377-388 (1995) lmaka Springer-Verlag 1995 Global identifiability for an inverse problem for the Schriidinger equation in a magnetic field Gen Nakamura l, Ziqi Sun2'*, Gunther Uhlmann3,** I Department of Mathematics, Science University of Tokyo, Shinjuku-ku,Tokyo 162, Japan 2 Department of Mathematics and Statistics, Wichita State University, Wichita, KS 67260, USA 3 Department of Mathematics, University of Washington, Seattle, WA 98195, USA Received: 20 December 1993 / Revised version: 11 July 1994 Mathematics Subject Classification (1991): 35 R, 35 Q 1 Introduction In this paper we study an inverse boundary value problem for the Schrrdinger equation in the presence of a magnetic potential. Let f2 be a bounded domain in R n, n ~ 3, with smooth boundary. The Schrrdinger equation in a magnetic potential is given by (1.1) H~,q= ~-~( 1 0 )2 J=~ 7~xi +Aj(x) +q(x), i = ~/-L-~, where X = (AbAE,...,An) E CI(I]) is the magnetic potential and q E L~(12) is the electric potential. The magnetic field is the rotation of the magnetic potential, rot(A). We assume that X and q are real-valued function and thus (1.1) is self- adjoint. We also assume that zero is not a Dirichlet eigenvalue of (1.1) on f2, so that the boundary value problem ( HX, qU = 0 in f2 (1.2) ulon= f ~/4 has a unique solution u E Hl(fl). The Dirichlet-to-Neumann map AZq which maps H into H- is defined by (1.3) AX, q : f --~ O~v v + i(X.v)f, f E H t3fl where u is the unique solution to (1.2), and v is the unit outer normal on Off. * Partially supported by NSF Grant DMS--9123742 ** Partially supported by NSF Grant DMS-9100178 and ONR grant N00014-93-1-0295