  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.
