  A waveguide coincides with a strip having two narrows of diameter $\epsilon$.
Electron motion is described by the Helmholtz equation with Dirichlet boundary
condition. The part of waveguide between the narrows plays the role of
resonator and there can occur electron resonant tunneling. This phenomenon
consists in the fact that, for an electron with energy $E$, the probability
$T(E)$ to pass from one part of the waveguide to the other part through the
resonator has a sharp peak at $E=E_{res}$, where $E_{res}$ denotes a "resonant"
energy. In the present paper, we compare the asymptotics of
$E_{res}=E_{res}(\epsilon)$ and $T(E)=T(E, \epsilon)$ as $\epsilon \to 0$ with
the corresponding numerical results obtained by approximate computing the
waveguide scattering matrix. We show that there exists a band of $\epsilon$
where the asymptotics and numerical results are in close agreement. The
numerical calculations become inefficient as $\epsilon$ decreases; however, at
such a condition the asymptotics remains reliable. On the other hand, the
asymptotics gives way to the numerical method as $\epsilon$ increases; in fact,
for wide narrows the resonant tunneling vanishes by itself.
  Though, in the present paper, we consider only a 2D waveguide, the
applicability of the methods goes far beyond the above simplest model. In
particular, the same approach will work for asymptotic and numerical analysis
of resonant tunneling in 3D quantum waveguides.
