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Uniform Spectral Properties of One-Dimensional Quasicrystals, II. The Lyapunov Exponent

Damanik, David and Lenz, Daniel (1999) Uniform Spectral Properties of One-Dimensional Quasicrystals, II. The Lyapunov Exponent. Letters in Mathematical Physics, 50 (4). pp. 245-257. ISSN 0377-9017. doi:10.1023/A:1007614218486.

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In this Letter we introduce a method that allows one to prove uniform local results for one-dimensional discrete Schrödinger operators with Sturmian potentials. We apply this method to the transfer matrices in order to study the Lyapunov exponent and the growth rate of eigenfunctions. This gives uniform vanishing of the Lyapunov exponent on the spectrum for all irrational rotation numbers. For irrational rotation numbers with bounded continued fraction expansion, it gives uniform existence of the Lyapunov exponent on the whole complex plane. Moreover, it yields uniform polynomial upper bounds on the growth rate of transfer matrices for irrational rotation numbers with bounded density. In particular, all our results apply to the Fibonacci case.

Item Type:Article
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URLURL TypeDescription Paper ReadCube access
Damanik, David0000-0001-5924-3849
Lenz, Daniel0000-0001-5820-475X
Additional Information:© Kluwer Academic Publishers 1999. (Received: 8 July 1999; revised version: 21 February 2000) D.D. was supported by the German Academic Exchange Service through Hochschulsonderprogramm III (Postdoktoranden) and D.L. received financial support from Studienstiftung des Deutschen Volkes (Doktorandenstipendium), both of which are gratefully acknowledged.
Funding AgencyGrant Number
Deutscher Akademischer Austauschdienst (DAAD)UNSPECIFIED
Studienstiftung des deutschen VolkesUNSPECIFIED
Subject Keywords:Schrödinger operators; quasiperiodic potentials; Lyapunov exponent
Issue or Number:4
Classification Code:Mathematics Subject Classi¢cations (2000): 81Q10, 47B80.
Record Number:CaltechAUTHORS:20170408-155954475
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Official Citation:Damanik, D. & Lenz, D. Letters in Mathematical Physics (1999) 50: 245. doi:10.1023/A:1007614218486
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:76108
Deposited By: 1Science Import
Deposited On:13 Jun 2017 19:28
Last Modified:15 Nov 2021 16:57

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