Quantum waveguides with a lateral semitransparent barrier: spectral and scattering properties

TL;DR

This paper analyzes the spectral and scattering properties of quantum waveguides with a semitransparent barrier, revealing conditions for bound states and numerically studying eigenvalues, eigenfunctions, and effects of geometry and magnetic fields.

Contribution

It introduces a model of quantum waveguides with a semitransparent barrier and investigates bound states, spectral properties, and effects of geometry and magnetic fields, using numerical and analytical methods.

Findings

Existence of bound states below the essential spectrum under attractive mean coupling.

Numerical determination of eigenvalues, eigenfunctions, and scattering matrix.

Insights into the emergence of ground-state energy and nodal line properties.

Abstract

We consider a quantum particle in a waveguide which consists of an infinite straight Dirichlet strip divided by a thin semitransparent barrier on a line parallel to the walls which is modeled by a $\delta$ potential. We show that if the coupling strength of the latter is modified locally, i.e. it reaches the same asymptotic value in both directions along the line, there is always a bound state below the bottom of the essential spectrum provided the effective coupling function is attractive in the mean. The eigenvalues and eigenfunctions, as well as the scattering matrix for energies above the threshold, are found numerically by the mode-matching technique. In particular, we discuss the rate at which the ground-state energy emerges from the continuum and properties of the nodal lines. Finally, we investigate a system with a modified geometry: an infinite cylindrical surface threaded by a homogeneous magnetic field parallel to the cylinder axis. The motion on the cylinder is again constrained by a semitransparent barrier imposed on a ``seam'' parallel to the axis.