The Kronig-Penney model in a quadratic channel with $\delta $ interactions. II : Scattering approach

TL;DR

This paper introduces a scattering approach to analyze the Kronig-Penney model with $\, ext{delta}"$ interactions in a quadratic channel, revealing new spectral behaviors and band structures, with numerical and semi-classical insights.

Contribution

It develops a scattering matrix method for the Kronig-Penney model, enabling analysis of spectral properties and band structures in a quadratic channel with $\, ext{delta}"$ interactions, extending previous work.

Findings

Spectral bands exhibit surprising behaviors under periodic conditions.

Numerical computation of spectral bands is feasible and informative.

Semi-classical analysis provides qualitative understanding of spectral features.

Abstract

The main purpose of the present paper is to introduce a scattering approach to the study of the Kronig-Penney model in a quadratic channel with $\delta$ interactions, which was discussed in full generality in the first paper of the present series. In particular, a secular equation whose zeros determine the spectrum will be written in terms of the scattering matrix from a single $\delta$. The advantages of this approach will be demonstrated in addressing the domain with total energy $E\in [0,\frac{1}{2})$, namely, the energy interval where, for under critical interaction strength, a discrete spectrum is known to exist for the single $\delta$ case. Extending this to the study of the periodic case reveals quite surprising behavior of the Floquet spectra and the corresponding spectral bands. The computation of these bands can be carried out numerically, and the main features can be qualitatively explained in terms of a semi-classical framework which is developed for the purpose.