Inverse Streutheorie für die Schrödingergleichung mit Magnetfeld

Silke Arians

ISBN 978-3-89722-028-7
80 pages, year of publication: 1998
price: 40.00 €

We consider the inverse scattering problem for a non-relativistic Hamiltonian of a quantum particle in an electromagnetic field in at least two space dimensions. Knowing the scattering data we reconstruct the Hamiltonian or the physical fields. For that purpose we study the scattering operator, which is an operator from the incoming to the outgoing asymptotic configurations, in the high velocity limit. We use that the translation of the wave packets dominates the scattering for very high velocities, which simplifies the time evolution. We show the uniqueness of the inverse scattering problem and give reconstruction formulae for the magnetic field and the electric potential.

We consider on one hand the case of a magnetic field decaying sufficiently fast at infinity and on the other hand the Hamiltonian of a quantum particle in a homogeneous magnetic field and an electric potential in space dimension three. We show that the high velocity limit of the scattering operator uniquely determines the magnetic field and the electric potential if the electric potential is of short range. If, in addition, long-range potentials are present, some knowledge of the long-range part is needed to define a modified Dollard wave operator and a scattering operator. Again its high velocity limit uniquely determines the magnetic field and the total electric potential. Moreover, we give explicit error bounds which are inverse proportional to the velocity. In addition, we analyze the freedom of gauge transformations. By means of an example we compare different gauges. We use this in order to weaken the decay assumptions on the vector potential for the proofs of the reconstruction formulae if the magnetic field is compactly supported. We generalize our results to a system of two interacting quantum particles with opposite electric charges.

The first order approximation of the high velocity limit of the scattering operators is the X-ray transform of the electric potential. Finally, we therefore consider the inversion of the X-ray and Radon transforms as well as the continuity of the transforms and their inverses. Keywords: X-ray transform; inverse scattering problem; non-relativistic Hamiltonian; quantum particle in an electromagnetic field; uniqueness; reconstruction formulae; high velocity limit; Dollard wave operator; scattering operator; gauge transformations; two interacting quantum particles; Radon transforms