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

null Smilansky

doi:10.1088/1751-8121/ac9357

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

Smilansky

Weizmann Institute

Research output: Contribution to journal › Article › peer-review

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 δ 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 δ. The advantages of this approach will be demonstrated in addressing the domain with total energy E ∈ [ 0 , 1 2 ) , namely, the energy interval where, for under critical interaction strength, a discrete spectrum is known to exist for the single δ 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.

Original language	English
Article number	424007
Journal	Journal of Physics A: Mathematical and Theoretical
Volume	55
Issue number	42
DOIs	https://doi.org/10.1088/1751-8121/ac9357
Publication status	Published - 28 Oct 2022
Externally published	Yes

Bibliographical note

Funding Information:
This work was concluded in the University of Bath, where the author was nominated a David Parkin visiting professor in the department of mathematical studies. Thanks for the kind hospitality are very much due. This work started as a collaboration with Professor Italo Guarneri, and his ideas and critical comments where essential throughout. Thanks Italo! Thanks are also due to my two old time colleagues and friends—Professor Sven Gnutzmann who accompanied this project with a study of multi-mode quantum graphs and offered invaluable suggestions, and Professor Gregory Berkolaiko for pointing out several errors which existed in the previous version of the manuscript.

Keywords

Floquet spectrum
Kronig-Penney model
multimode graph
quantum graphs

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Statistics and Probability
Modelling and Simulation
Mathematical Physics
General Physics and Astronomy

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10.1088/1751-8121/ac9357Licence: CC BY

Cite this

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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 δ 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 δ. The advantages of this approach will be demonstrated in addressing the domain with total energy E ∈ [ 0 , 1 2 ) , namely, the energy interval where, for under critical interaction strength, a discrete spectrum is known to exist for the single δ 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.",

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N2 - 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 δ 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 δ. The advantages of this approach will be demonstrated in addressing the domain with total energy E ∈ [ 0 , 1 2 ) , namely, the energy interval where, for under critical interaction strength, a discrete spectrum is known to exist for the single δ 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.

AB - 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 δ 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 δ. The advantages of this approach will be demonstrated in addressing the domain with total energy E ∈ [ 0 , 1 2 ) , namely, the energy interval where, for under critical interaction strength, a discrete spectrum is known to exist for the single δ 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.

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KW - Kronig-Penney model

KW - multimode graph

KW - quantum graphs

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

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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