Asymptotic and numerical analysis of slowly varying two-dimensional quantum waveguides

The work is devoted to the asymptotic and numerical analysis of the wave function propagating in two-dimensional quantum waveguides with confining potentials supported on slowly varying tubes. The leading term of the asymptotics of the wave function is determined by an adiabatic approach and the WKB approximation. Unlike other similar studies, in the present work we consider arbitrary bounded potentials and obtain exact solutions for the thresholds, and for the transverse modes in the form of power series of the spectral parameter. Our approach leads to an effective numerical method for the analysis of such quantum waveguides and for the tunnel effect observed in sections of the waveguide that shrink or widen too much. Several examples of interest show the applicability of the method.

Резонансы в двумерных квантовых волноводах, разделенных полупрозрачным барьером с малым отверстием

A pair of coupled quantum waveguides with a common semitransparent boundary having a small aperture is considered, which leads to resonance phenomena during the passage of an electron. Using the method of matching asymptotic expansions, formulas are obtained for the first terms in the expansions of resonances (quasi-eigenvalues) and resonance states. The influence of the geometry of the system and the transparency parameter of the boundary on the lifetime of resonance states is investigated