Asymptotic and numerical studies of resonant tunneling in 2D quantum waveguides of variable cross-section

Abstract: 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.