Vol. 3, No. 2, 2022

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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

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.

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