Inverse scattering problem for the Schrödinger equation with magnetic potential at a fixed energy

Abstract

In this article we consider the Schrδdinger operator in R n ,n ^ 3, with electric and magnetic potentials which decay exponentially as \x\ -> oo. We show that the scattering amplitude at fixed positive energy determines the electric potential and the magnetic field.

Determining a Magnetic Schrödinger Operator from Partial Cauchy Data

“…The latter was studied under various regularity assumptions on the magnetic and electrical potentials in [9], for small compactly supported magnetic potential and compactly supported electric potential. This result was extended in [3] for exponentially decaying magnetic and electric potentials with no smallness assumption.…”

Semiclassical Pseudodifferential Calculus and the Reconstruction of a Magnetic Field

“…Lemma 4.1 is a global nonsmooth version of the pseudodifferential conjugation technique in [16] (see also [17]). Similar ideas have been used in inverse scattering [5], [8], nonlinear Schrödinger equations [26], [9] and periodic Schrödinger operators [22]. The problem in extending the method to the global case is seen in (15), where the derivatives in ξ of the symbol grow in x.…”

Identifiability at the boundary for first-order terms

“…The problem of identifiability of a vector potential was first studied by Sun [11]. Other authors who have considered this problem include Eskin and Ralston [5] and Tolmasky [13]. In his Ph.D. thesis [10], Salo gives recovery of a continuous vector potential in domains that are C 1,1 .…”