Bound states and metastability near a scatterer in crossed electromagnetic fields

The eigenvalue problem of an electron in the plane in the presence of a repulsive scatterer is studied. The electron is subject to a weak in-plane electric field and a magnetic field perpendicular to the plane. The associated magnetic length is much larger than the range of the scatterer. In this parameter region it is natural to follow Prange and treat the scatterer basically as a repulsive δ-function. However, the finite range of the scatterer is essential in that it provides the cutoffs necessary to make the problem mathematically well posed. We demonstrate that a true δ-function is unable to trap an electron in a finite electric field, no matter how small. At high Landau levels we find semi-quantitative agreement with recent classical results on electron trapping. With sharp cutoffs one bound state per Landau level is found for sufficiently weak electric fields. As the strength of the electric field is increased, the role of the bound state is taken over by a metastable wave packet which remains close to the scatterer for an exceedingly long time. This wave packet is explicitly constructed. With smooth cutoffs, all bound states become submerged in the continuum, and only long-lived wavepackets remain.