On the spectrum of the one-dimensional Schrödinger operator

Nguyen Dinh Cong; Roberta Fabbri

doi:10.3934/dcdsb.2008.9.541

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2008, Volume 9, Issue 3&4: 541-554. Doi: 10.3934/dcdsb.2008.9.541

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On the spectrum of the one-dimensional Schrödinger operator

Nguyen Dinh Cong^1, and
Roberta Fabbri^2,

1.
Institute of Mathematics, Vietnamese Academy of Science and Technology, 18 Hoang Quoc Viet Road, 10307 Hanoi
2.
Università di Firenze, Dipartimento di Sistemi e Informatica, Via Santa Marta 3, 50139 Firenze

Received: December 31, 2006

Revised: June 30, 2007

Published: April 2008

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Abstract

The spectral theory of the one-dimensional Schrödinger operator with a quasi-periodic potential can be fruitfully studied considering the corresponding differential system. In fact the presence of an exponential dichotomy for the system is equivalent to the statement that the energy $E$ belongs to the resolvent of the operator. Starting from results already obtained for the spectrum in the continuous case, we show that in the discrete case a generic bounded measurable Schrödinger cocycle has Cantor spectrum.

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Mathematics Subject Classification: Primary: 34D08, 34D09; Secondary: 37B55.

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