Model tunneling problems in a high magnetic field

We have studied simple tunneling problems in two dimensions in the presence of a high transverse magnetic field both by numerical integration of the Schrödinger equation and by semiclassical evaluation of the path integral. We have chosen three model potentials: (i) asymmetric single well, (ii) symmetric double well, and (iii) quadruple well. We find that the semiclassical approach is analytically tractable and gives a very accurate description of the exponential and oscillatory behaviors of the tunneling matrix elements. A precise definition of the Aharonov-Bohm phase for the tunneling paths is given. In addition to the Aharonov-Bohm phase, there is also a geometrical phase coming from the fluctuation determinant, and we find that for every closed loop it is exactly.