Asymptotics of resonant tunneling in a two-dimensional quantum waveguide with several equal resonators

ABSTRACT

The domain occupied by the waveguide is a strip with n+1 equal narrows of diameter ϵ. The wave function of a free electron satisfies the Dirichlet boundary value problem for the Helmholtz equation. Any part of the waveguide between two neighboring narrows plays the role of a resonator. Near a simple eigenvalue of the closed resonator, there are n resonant peaks of height close to 1. We let $\epsilon \to 0$ and obtain asymptotic formulas for the resonant values of the spectral parameter and for the widths of the resonant peaks at their half-height. The behavior of the transmission coefficient in a neighborhood of a resonance is described.

KEYWORDS:

• Quantum waveguide
• variable cross-section
• the Helmholtz equation
• resonant tunneling
• asymptotic description

AMS Subject Classifications:

• 35Q40
• 35B40
• 35B34

Additional information

Funding

This work was supported by Russian Science Foundation [grant number 17-11-01126].