Semiclassical Pseudodifferential Calculus and the Reconstruction of a Magnetic Field

Abstract

We give a procedure for reconstructing a magnetic field and electric potential from boundary measurements given by the Dirichlet to Neumann map for the magnetic Schrödinger operator in R n , n ≥ 3. The magnetic potential is assumed to be continuous with L ∞ divergence and zero boundary values. The method is based on semiclassical pseudodifferential calculus and the construction of complex geometrical optics solutions in weighted Sobolev spaces.

Determining a Magnetic Schrödinger Operator from Partial Cauchy Data

“…The regularity assumption on the magnetic potential was improved in [13] to C 2/3+ǫ , ǫ > 0, and to Dini continuous in [10]. Recently in [11] a method was given for reconstructing the magnetic field and the electric potential under some regularity assumptions on the magnetic potential.…”

Determining nonsmooth first order terms from partial boundary measurements

“…The regularity of W was relaxed to C 1 in [21] and to Dini continuous in [18]. A constructive procedure for recovering dW and V from N W,V is given in [17]. The related inverse scattering problem has been studied in [7].…”

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

“…We also refer to Bukhgeim and Uhlamnn [12], Hech-Wang [21], Salo [36] and Uhlmann [44] as a survey. In [16] Dos Santos Ferreira, Kenig, Sjostrand, Uhlmann prove that the knowledge of the Cauchy data for the Schrödinger equation in the presence of magnetic potential, measured on possibly very small subset of the boundary, determines uniquely the magnetic field.…”