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Uniform Spectral Properties of One-Dimensional Quasicrystals, III. α-Continuity

Damanik, David and Killip, Rowan and Lenz, Daniel (2000) Uniform Spectral Properties of One-Dimensional Quasicrystals, III. α-Continuity. Communications in Mathematical Physics, 212 (1). pp. 191-204. ISSN 0010-3616. doi:10.1007/s002200000203.

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We study the spectral properties of one-dimensional whole-line Schrödinger operators, especially those with Sturmian potentials. Building upon the Jitomirskaya–Last extension of the Gilbert–Pearson theory of subordinacy, we demonstrate how to establish α-continuity of a whole-line operator from power-law bounds on the solutions on a half-line. However, we require that these bounds hold uniformly in the boundary condition. We are able to prove these bounds for Sturmian potentials with rotation numbers of bounded density and arbitrary coupling constant. From this we establish purely α-continuous spectrum uniformly for all phases. Our analysis also permits us to prove that the point spectrum is empty for all Sturmian potentials.

Item Type:Article
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URLURL TypeDescription Paper ReadCube access
Damanik, David0000-0001-5924-3849
Killip, Rowan0000-0002-4272-7916
Lenz, Daniel0000-0001-5820-475X
Additional Information:© Springer-Verlag Berlin Heidelberg 2000. Received: 29 September 1999 / Accepted: 14 January 2000 Communicated by B. Simon. D. D. was supported by the German Academic Exchange Service through Hochschulsonderprogramm III (Postdoktoranden), R. K. was supported, in part, by an Alfred P. Sloan Doctoral Dissertation Fellowship, and D. L. received financial support from Studienstiftung des Deutschen Volkes (Doktorandenstipendium), all of which are gratefully acknowledged.
Funding AgencyGrant Number
Deutscher Akademischer Austauschdienst (DAAD)UNSPECIFIED
Alfred P. Sloan FoundationUNSPECIFIED
Studienstiftung des deutschen VolkesUNSPECIFIED
Issue or Number:1
Record Number:CaltechAUTHORS:20170408-171841154
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Official Citation:Damanik, D., Killip, R. & Lenz, D. Comm Math Phys (2000) 212: 191.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:76322
Deposited By: 1Science Import
Deposited On:13 Mar 2018 18:24
Last Modified:15 Nov 2021 16:59

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