Effect of magnetic field on resonant tunneling in 3D waveguides of variable cross-section

We consider an infinite three-dimensional waveguide that far from the coordinate origin coincides with a cylinder. The waveguide has two narrows of diameter $\epsilon$. The narrows play the role of effective potential barriers for the longitudinal electron motion. The part of waveguide between the narrows becomes a “resonator”, and there can arise conditions for electron resonant tunneling. A magnetic field in the resonator can change the basic characteristics of this phenomenon. In the presence of a magnetic field, the tunneling phenomenon is feasible for producing spin-polarized electron flows consisting of electrons with spins of the same direction. We assume that the whole domain occupied by a magnetic field is in the resonator. An electron wave function satisfies the Pauli equation in the waveguide and vanishes at its boundary. Taking $\epsilon$ as a small parameter, we derive asymptotics for the probability $T(E)$ of an electron with energy $E$ to pass through the resonator, for the “resonant energy”,$E_{res}$, where $T(E)$ takes its maximal value, and for some other resonant tunneling characteristics.