  We study two-dimensional magnetic Schr\"odinger operators with a magnetic
field that is equal to b>0 for x > 0 and (-b) for x < 0. This magnetic
Schr\"odinger operator exhibits a magnetic barrier at x=0. The unperturbed
system is invariant with respect to translations in the y-direction. As a
result, the Schr\"odinger operator admits a direct integral decomposition. We
analyze the band functions of the fiber operators as functions of the wave
number and establish their asymptotic behavior. Because the fiber operators are
reflection symmetric, the band functions may be classified as odd or even. The
odd band functions have a unique absolute minimum. We calculate the effective
mass at the minimum and prove that it is positive. The even band functions are
monotone decreasing. We prove that the eigenvalues of an Airy operator,
respectively, harmonic oscillator operator, describe the asymptotic behavior of
the band functions for large negative, respectively positive, wave numbers. We
prove a Mourre estimate for perturbations of the magnetic Schr\"odinger
operator and establish the existence of absolutely continuous spectrum in
certain energy intervals. We prove lower bounds on magnetic edge currents for
states with energies in the same intervals. We also prove that these lower
bounds imply stable lower bounds for the asymptotic currents. We study the
perturbation by slowly decaying negative potentials and establish the
asymptotic behavior of the eigenvalue counting function for the infinitely-many
eigenvalues below the bottom of the essential spectrum.
