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

Abstract

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.

Quantum soft waveguides with resonances induced by broken symmetry

“…(2.6) 1 We will use the notation I for the identity operator regardless of the space in which it acts.…”

“…In [10], it has been shown that for a weak deformation the existence of the discrete spectrum points depends on the total area of deformation which is assumed to be positive if the deformation is localized on Σ \ Ω and negative if the deformation is situated on Ω \ Σ. It is also worth mentioning the paper [1], where the authors have considered two dimensional quantum waveguides modelled by confining potentials supported on slowly varying strips. The asymptotic and numerical analysis of the wave functions propagating in such waveguides has been proceeded and, for example, the leading asymptotic term of the wave functions has been determined by the adiabatic approach and the WKB approximation.…”