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On the spectrum of the Kronig-Penney model in a constant electric field

Frank, Rupert L. and Larson, Simon (2021) On the spectrum of the Kronig-Penney model in a constant electric field. . (Unpublished)

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We are interested in the nature of the spectrum of the one-dimensional Schrödinger operator −d²/dx² − Fx + ∑_(n∈ℤ) g_nδ(x−n)in L²(ℝ) with F > 0 and two different choices of the coupling constants {g_n}n ∈ ℤ. In the first model g² ≡ λ and we prove that if F ∈ π²ℚ then the spectrum is ℝ and is furthermore absolutely continuous away from an explicit discrete set of points. In the second model g_n 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 ℝ and it is dense pure point if F < λ²/2 and purely singular continuous if F > λ²/2.

Item Type:Report or Paper (Discussion Paper)
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URLURL TypeDescription Paper
Frank, Rupert L.0000-0001-7973-4688
Larson, Simon0000-0002-0057-8211
Additional Information:© 2021 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. U.S. National Science Foundation grants DMS-1363432 and DMS-1954995 (R.L.F.) and Knut and Alice Wallenberg Foundation grant KAW 2018.0281 (S.L.) are acknowledged.
Funding AgencyGrant Number
Knut and Alice Wallenberg Foundation2018.0281
Record Number:CaltechAUTHORS:20211004-232828060
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:111207
Deposited By: George Porter
Deposited On:07 Oct 2021 19:14
Last Modified:07 Oct 2021 19:14

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