1,061 research outputs found
High-Velocity Estimates for Schr\"odinger Operators in Two Dimensions: Long-Range Magnetic Potentials and Time-Dependent Inverse Scattering
We introduce a general class of long-range magnetic potentials and derive
high velocity limits for the scattering operators in quantum mechanics, in the
case of two dimensions. We analyze the high velocity limits in the presence of
an obstacle and we uniquely reconstruct from them the electric potential and
the magnetic field outside the obstacle. We also reconstruct the inaccessible
magnetic fluxes produced by fields inside the obstacle modulo . For
every magnetic potential in our class we prove that its behavior at
infinity () can be
characterized in a natural way. Under very general assumptions we prove that
can be uniquely
reconstructed for every . We characterize
properties of the support of the magnetic field outside the obstacle that
permit us to uniquely reconstruct either for all or for in a subset of
. We also give a wide class of magnetic fields outside the
obstacle allowing us to uniquely reconstruct the total magnetic flux (and
for all ). This
is relevant because, as it is well-known, in general the scattering operator
(even if is known for all velocities or energies) does not define uniquely the
total magnetic flux (and ). We analyze additionally
injectivity (i.e., uniqueness without giving a method for reconstruction) of
the high velocity limits of the scattering operator with respect to
. Assuming that the magnetic field outside the
obstacle is not identically zero, we provide a class of magnetic potentials for
which injectivity is valid
- β¦