Vol. 3, No. 2, 2022

Download this article
Download this article For screen
For printing
Recent Issues
Volume 5, Issue 1
Volume 4, Issue 4
Volume 4, Issue 3
Volume 4, Issue 2
Volume 4, Issue 1
Volume 3, Issue 4
Volume 3, Issue 3
Volume 3, Issue 2
Volume 3, Issue 1
Volume 2, Issue 4
Volume 2, Issue 3
Volume 2, Issue 2
Volume 2, Issue 1
Volume 1, Issue 1
The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
 
Subscriptions
 
ISSN (electronic): 2690-1005
ISSN (print): 2690-0998
Author Index
To Appear
 
Other MSP Journals
This article is available for purchase or by subscription. See below.
On the spectrum of the Kronig–Penney model in a constant electric field

Rupert L. Frank and Simon Larson

Vol. 3 (2022), No. 2, 431–490
Abstract

We are interested in the nature of the spectrum of the one-dimensional Schrödinger operator

d2 dx2 Fx + ngnδ(x n) in  L2()

with F > 0 and two different choices of the coupling constants {gn}n. In the first model, gn λ, and we prove that if F π2 then the spectrum is and is furthermore absolutely continuous away from an explicit discrete set of points. In the second model, the gn are independent random variables with mean zero and variance λ2. Under certain assumptions on the distribution of these random variables, we prove that almost surely the spectrum is and it is dense pure point if F < 1 2λ2 and purely singular continuous if F > 1 2λ2.

PDF Access Denied

We have not been able to recognize your IP address 3.238.112.198 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

Keywords
spectral theory, one-dimensional Schrödinger operators, Kronig–Penney, random Schrödinger operators
Mathematical Subject Classification
Primary: 34L40, 47B80, 81Q10
Milestones
Received: 22 May 2021
Revised: 10 November 2021
Accepted: 16 December 2021
Published: 8 July 2022
Authors
Rupert L. Frank
Mathematisches Institut
Ludwig-Maximilians-Universität München
Munich
Germany
Munich Center for Quantum Science and Technology
Munich
Germany
Department of Mathematics
California Institute of Technology
Pasadena, CA
United States
Simon Larson
Chalmers University of Technology
Gothenburg
Sweden
University of Gothenburg
Gothenburg
Sweden