Exact quantum-mechanical propagator for an electron in a saddle-point potential and a magnetic field

We study the time-dependent propagator for an electron which moves in a plane and which is subject to quadrupolar electric fields with certain reflection symmetries, and to a uniform magnetic field perpendicular to the plane. It is shown that the electric potential, which has a saddle point at the origin, can be produced by voltages applied to four symmetrically configured cylindrical electrodes whose axes are perpendicular to the plane. We use path-integral methods to derive a closed-form expression for the propagator, and comment in some detail on the structure of its normalization factor.