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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 $\varepsilon$. 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 $\varepsilon$ 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.