Asymptotics of the Resonant Tunneling of High-Energy Electrons in Two-Dimensional Quantum Waveguides of Variable Cross-Section

• Published in: Journal of mathematical sciences (New York, N.Y.) Vol. 238; no. 5; pp. 736 - 749
• Main Author: Sarafanov, O. V.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.05.2019; Springer; Springer Nature B.V
• Subjects: Asymptotic properties; ASYMPTOTIC SOLUTIONS; Bibliographies; BOUNDARY CONDITIONS; CROSS SECTIONS; DIRICHLET PROBLEM; EIGENVALUES; Electrons; EQUATIONS; Helmholtz equations; High energy electrons; MATHEMATICAL METHODS AND COMPUTING; Mathematics; Mathematics and Statistics; PEAKS; Resonant tunneling; Resonators; TUNNEL EFFECT; TWO-DIMENSIONAL CALCULATIONS; WAVE FUNCTIONS; WAVEGUIDES
• ISSN: 1072-3374; 1573-8795
• DOI: 10.1007/s10958-019-04271-4

A waveguide occupies a strip in ℝ 2 having two identical narrows of small diameter ε . 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 ε → 0, a neighborhood of a narrow is assumed to transform into a neighborhood of the common vertex of two vertical angles. The part of the waveguide between the narrows as ε = 0 is called the resonator. An asymptotics of the transmission coefficient is obtained in the waveguide as ε → 0. Near a degenerate eigenvalue of the resonator, the leading term of the asymptotics has two sharp peaks. Positions and shapes of the resonant peaks are described.