Point interactions in a strip

TL;DR

This paper investigates the spectral and scattering effects of finite and infinite point interactions in a quantum strip, revealing bound states, resonances, and spectral gaps, with extensions to related geometries and random perturbations.

Contribution

It provides a detailed analysis of point interactions in a quantum strip, including spectral properties, resonances, and effects of periodic and random arrangements, extending known results to new geometries.

Findings

Existence of discrete eigenvalues due to point interactions.

Infinite series of resonances approaching transverse mode thresholds.

Spectral gaps can form with periodic point perturbations.

Abstract

We study the behavior of a quantum particle confined to a hard--wall strip of a constant width in which there is a finite number $ N $ of point perturbations. Constructing the resolvent of the corresponding Hamiltonian by means of Krein's formula, we analyze its spectral and scattering properties. The bound state--problem is analogous to that of point interactions in the plane: since a two--dimensional point interaction is never repulsive, there are $ m $ discrete eigenvalues, $ 1\le m\le N $, the lowest of which is nondegenerate. On the other hand, due to the presence of the boundary the point interactions give rise to infinite series of resonances; if the coupling is weak they approach the thresholds of higher transverse modes. We derive also spectral and scattering properties for point perturbations in several related models: a cylindrical surface, both of a finite and infinite heigth, threaded by a magnetic flux, and a straight strip which supports a potential independent of the transverse coordinate. As for strips with an infinite number of point perturbations, we restrict ourselves to the situation when the latter are arranged periodically; we show that in distinction to the case of a point--perturbation array in the plane, the spectrum may exhibit any finite number of gaps. Finally, we study numerically conductance fluctuations in case of random point perturbations.