Comparison of asymptotic and numerical approaches to the study of the resonant tunneling in a two-dimensional symmetric quantum waveguide of variable cross-section

The waveguide coincides with a strip having two narrows of width $\varepsilon$. An electron wave function satisfies the Dirichlet boundary value problem for the Helmholtz equation. The part of the waveguide between the narrows serves as a resonator and conditions for the electron resonant tunneling can occur. In the paper, asymptotic formulas as $\varepsilon\to0$ for characteristics of the resonant tunneling are used. The asymptotic results are compared with numerical ones obtained with approximate calculation of the scattering matrix for energies in the interval between the second and the third thresholds. The comparison allows to state an interval of $\varepsilon$, where the asymptotic and numerical approaches agree. The suggested methods can be applied to more complicated models than one considered in the paper. In particular, the same approach can be used for asymptotic and numerical analysis of the tunneling in three-dimensional quantum waveguides of variable cross-section.