Inverse Scattering Problems For Schrödinger Operators with Magnetic and Electric Potentials

Abstract

Consider the Schrödinger equation in R ⁿ, n ≥ 3, with magnetic potential A(x) = (A ₁(x),…,A _n (x)) and electric potential V(x):

$$ {\left( {\frac{1}{i}\frac{\partial }{{\partial x}} + A\left( x \right)} \right)^{2}}u + V\left( x \right)u = {k^{2}}u$$

(1.1)

or equivalently

$$ - \Delta u - 2i\sum\limits_{{j = 1}}^{n} {Aj} \left( x \right)\frac{{\partial u}}{{\partial {x_{j}}}} + q\left( x \right)u = {k^{2}}u$$

(1.2)

where We will assume that the potentials A and V are real-valued and exponentially decreasing, i.e.