We study the effects of tunneling between the classical trajectories of a two-dimensional electron in a strong magnetic field and a slowly varying external potential on the quantum energy levels of the electron and on the phonon-induced hopping rate of localized states. The tunneling is important when the classical orbits pass within a few magnetic lengths of a saddle point of the potential. We discuss first the case of a double-well potential with a single saddle point. For energies well below the potential energy at the saddle point, the spectrum is of the form one expects for two wells very distant from one another. For energies well above the energy at the saddle point, the spectrum is similar to that of a single-well potential. The spectrum evolves smoothly from the former to the latter in the energy interval between these two limits. In addition, we study the spectrum for orbits coupled at two saddle points, and show that our technique can be applied to any number of orbits coupled at any number of saddle points. The method may be used to find the energy spectrum of an electron in a slowly varying random potential. We also calculate the tunnel splitting of states of a symmetric double well, and the phonon-induced hopping rate between localized states of an asymmetric double well. In both cases, we find that these quantities fall off as Gaussians in the distance of closest approach of the classical trajectories to the saddle point.

• Received 19 January 1988

DOI:https://doi.org/10.1103/PhysRevB.38.996