Asymptotics of the resonant tunneling of high-energy electrons in two-dimensional quantum waveguides of variable cross-section

The waveguide occupies a strip in $\mathbb R^2$ having two identical narrows of small diameter $\varepsilon$. An electron wave function satisfies the Helmholtz equation with the homogeneous Dirichlet boundary condition. The energy of electrons may be rather high, i.e. any (fixed) number of waves can propagate in the strip far from the narrows. As $\varepsilon\to0$, a neighbourhood of a narrow is supposed to transform into a neighbourhood of the common vertex of two vertical angles. The part of the waveguide between the narrows as $\varepsilon=0$ is called the resonator. An asymptotics of the transition coefficient is obtained in the waveguide as $\varepsilon\to0$. Near a degenerate eigenvalue of the resonator, the leading term of the asymptotics has two sharp peaks. Position and shape of the resonant peaks are described.